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LETTERS   ON   LOGIC 

TO  A  YOUNG  MAN 
WITHOUT  A  MASTER 


BY 


HENRY  BRADFORD  SMITH 


THE  COLLEGE  BOOK  STORE 

I.A\t>  AVBNtfl 

pi.     .jEi.pui \,  i\\. 


•  ••_•.     • 

•        •   •      •      *  • 


LETTERS   ON   LOGIC 

TO  A  YOUNG  MAN 
WITHOUT  A  MASTER 


BY 


HENRY  BRADFORD  SMITH 


THE  COLLEGE  BOOK   STORE 

)435   WOODLAND   A\i  S\  1 
PHI]   \DELPHIA.   PA. 

1 920 


.  •  •  •  • 

• !  •    •  •  •   •  • . 


•  • 


•  .  *  •    •    ■  ■  ■ 


% 


OS   &b 


PRESS  OF 

THE  KEW  ERA  PRINTING  COMPANY 

LANCASTER,  PA. 


PREFATORY   NOTE 

Somewhat  more  than  a  year  ago  Professor  Edgar  A.  Singer,  Jr., 
asked  me  to  undertake  the  writing  of  a  text-book  for  use  in  the 
introductory  course  in  logic  at  the  University  of  Pennsylvania, 
placing  at  my  disposal  the  syllabus  of  a  course  of  lectures,  which 
he  had  o  to  give  some  years  before,  and  intimating  thai   I 

might  use  his  materials  in  any  way  that  I  saw  fit.  Tin-  present 
work  is  an  effort,  all  too  imperfect  as  I  am  aware,  to  meel  his 
request.  The  syllabus  itself  in  its  original  form  is  placed  at  tin- 
end  of  the  hook. 

I  shall  make  no  apology  for  the  changes,  which  have  been 
introduced  into  the  text.  They  invariably  involve  the  sacrifice 
of  a  proper  or  methodical  arrangement  in  order  to  gain  a  peda- 
gogical advantage.  And,  finally,  in  fairness  to  Professor  Singer 
it  must  be  understood  that  the  chapters  of  the  book  bear  no  direct 
relation  to  his  own  formal  lectures.  They  have  been  suggested 
by  the  outline  alone. 

The  comparative  brevity  of  the  work  is  explained  by  the  nature 
of  my  task.  It  would  have  been  risking  too  much  to  introduce 
historical  comments  of  my  own  or  to  venture  upon  questions  of 
application,  when  my  purpose  is  only  to  furnish  an  approach  in 
popular  form  to  a  very  original  contribution  to  logical  theory. 
Those  who  use  the  book  in  the  class  room  will  be  able  to  remedy 
this  defect  in  their  own  way.  The  syllabus  itself  will  be  found 
indispensable  for  those  who  intend  to  pursue  the  theory  still 
further;  in  particular,  for  those  who  de-ire  to  express  the  whole 
in  the  mor  at  form  of  a  symbolic  technique. 

H.  B.  S. 


583005 


LETTERS   ON    LOGIC 


I 

My  dear  Sir: 

Some  years  have  now  passed  since  you  first  wrote  to  me, 
modestly  describing  the  desires  of  an  intellect  deprived  from  the 
first  of  any  adequate  training  and  inquiring  how  such  defects 
as  your  mind  possessed  might  best  be  remedied.  I  recall  that 
you  had  begun  your  effort  for  self-improvement  in  the  field  of 
literature  but  that  you  had  found  such  studies  to  yield  a  pleasure, 
which  charmed  rather  than  satisfied  a  spirit  set  upon  overcoming 
obstacles.  It  was  natural,  then,  that  you  should  have  turned 
next  to  philosophy  and  certainly  fortunate  that  Plato  should 
have  lmii  the  first  writer  to  altogether  engage  your  attention. 
I  recollect  now,  however,  that  the  event  in  your  life,  to  which 
you  attached  a  prime  importance,  was  the  chance  discovery  of 
a  copy  of  Euclid's  Elements,  and  it  was  at  the  period,  when  you 
were  absorbed  in  reflections  of  this  genre,  when  you  had  become 
aware  that  mathematics  is  the  key  to  so  many  shrines,  that  you 
had  written  to  know  if  I  could  instruct  you  in  the  infinitesimal 
calculus.     It  was  here  that  our  correspondence  had  begun. 

I  have  often  turned  over  in  retrospect  these  and  other  details 
of  your  career  in  the  time  that  has  elapsed,  since  the  serious  turn 
in  your  financial  affairs  has  compelled  you  for  the  moment  to 
more  practical  preoccupations.  You  had  been  good  enough  to 
intimate  that  you  had  profited  by  that  course  of  instruction 
conveyed  to  you  by  Utter  and  that  you  had  profited  was  clearly 
shown  by  the  questions,  which  you  raised.  Hut,  then,  courtesy 
would  have  required  you  to  say  as  much.  The  proof  was  forth- 
coming only  yesterday  and  I  confess  that  I  feel  flattered  that 
you  should  now  a^k  me  to  introduce  you  to  another  science,  no 
less  engrossing  in  itself  and  whose  tradition  is  still  more  venerable. 

That   I  should  agree  to  undertake  what  you  so  delicately  u: 
you   yourself  of  all    men    must    have    been    the   last    to   doubt. 
Surely   none   i-   unaware    that    those,   whose  profession   it   is   to 
impart  knowledge  have  rarely  to  be  urged   to  exercise  what 

I 


2  Letters  on  Logic  to  a 

talents  they  possess;  that  they  often  act  upon  the  slenderest  of 
pretexts,  so  that  wise  men  will  not  provoke  their  comment  unless 
the  occasion  be  grave  and  the  need  of  their  advice  be  urgent. 
Need  I  add,  then,  that  I  accede  to  your  request,  nay,  that  I 
have  already  in  mind  a  plan  of  presentation? 

It  will  not  be  necessary  to  remind  a  reader  of  the  Dialogues, 
that  the  order  of  ideas,  which  one  follows,  when  he  begins  to 
apprehend  a  science,  is  in  no  wise  the  order,  in  which  those 
ideas  find  themselves  arranged,  when  that  study  has  made  a 
certain  progress;  that,  consequently,  the  pedagogical  order,  in 
which  the  matter  of  science  ought  to  be  arranged,  is  rather  the 
reverse  of  the  methodical.  The  latter  in  its  perfection  pertains 
only  to  God,  who  geometrizes  eternally;  the  former  is  a  necessity 
of  those  imperfections  to  which  the  flesh  is  heir.  In  the  logical 
ordering  of  the  materials  of  thought  those  accidents,  which  had 
come  into  being  with  the  materials  themselves  and  which  had 
become  attached  like  barnacles  to  the  proper  subject-matter, 
have  been  removed;  conditions,  which  at  first  appeared  as 
necessities  but  had  proved  redundant,  have  been  erased.  It  will 
furnish  you  no  surprise,  therefore,  if  our  plan  should  be  to  intro- 
duce such  and  such  a  condition  only  to  remove  it  later  on,  or  to 
postulate  the  truth  of  such  and  such  a  proposition  only  to  discover 
that  its  truth  may  be  made  to  depend  on  others  which  enter  at 
a  future  date.  I  shall  try  with  what  art  I  possess  to  lay  before 
you  the  drama  of  a  mind  at  work,  rather  than  reveal  at  the 
outset  the  finished  system,  which  emerges  for  the  first  time, 
when  that  work  is  done. 

It  only  remains  to  state  that  we  shall  not  be  able  to  concern 
ourselves  with  the  applications  of  the  science.  Such  matters  are 
of  the  most  delicate  sort  and  presuppose  always  the  highest 
degree  of  skill.  You  will  find  them  treated  in  the  current  texts. 
In  any  case  you  will  wish  to  bring  to  their  consideration  an 
adequate  theory  and  it  is  this  theory  alone,  which  we  propose 
to  supply.  We  shall  not  pretend  that  our  doctrine  is  complete 
but  we  shall  develop  it  so  far  as  is  usual  in  the  university  in  a 
first  semester  course. 


\'<>l  v.    M  \\    w  I  Him   i     \    MaSI  BR 


It  will  hardly  be  necessary  to  inform  a  student  of  mathematics 
of  the  difference  between  defining  an  object  <>r  relationship  by 
the  aid  of  synonyms,  after  the  fashion  of  the  dictionary,  and 

defining  it  by  means  of  its  formal  properties,  which  is  the  pro- 
cedure  of  science.  The  relations,  which  arc  the  proper  subject- 
matter  of  a  given  compartment  of  thought,  are  commonly  taken 
to  be  defined  by  the  set  of  axioms,  necessary  and  sufficient  to 
yield  all  the  theorems  characteristic  of  the  domain  of  application, 
which  is  in  question.  One  or  more  of  the  axioms,  which  taken 
together  determine  unambiguously  a  given  field  of  interpretation, 
may  hold  true  separately  of  other  relationships  than  the  one-, 
which  we  are  seeking  to  define. 

Suppose,  by  way  of  illustration,  that  we  wished  to  differentiate 
three  relations,  which  we  will  select  as  parallelism,  perpendicu- 
larity and  implication.  Two  axioms,  known  as  the  axiom  of 
transitivity  and  the  axiom  of  reciprocity,  will  then  be  enough  to 
effect  our  purpose.  If  all  three  of  our  relations  be  expressed  as 
transitive  we  should  have: 

(i )   If  x  is  parallel  to  y  and  y  is  parallel  to  z,  then  x  is  parallel 
to  Z\ 

(2)  If  .v  is  perpendicular  to  y  and  y  is  perpendicular  to  z, 

then  x  is  perpendicular  to  z; 

(3)  If  x  implies  y  and  y  implies  s,  then  x  implies  Z. 

<  )f  these  three  propositions  the  first  and  the  last  are  true  and  the 
ond  is  false.     If  our  three  relations  be  represented  as  recip- 
rocal, we  should  write: 

(i)    If  X  is  parallel  to  y.  then  y  is  parallel  to  x\ 
2)  If  x  is  perpendicular  to  y.  then  y  is  perpendicular  to  x\ 

\  If  X  implies  y.  then  y  implies  X. 
Ihre  the  first  and  the  second  propositions  are  true  and  the 
third  i-  false.  Consequently,  parallelism  is  both  transitive  and 
reciprocal,  perpendicularity  is  reciprocal  but  not  transitive  and 
implication  is  transitive  but  not  reciprocal.  If  it  were  only 
desired  to  distinguish  between  these  three,  enough  <>f  their 
properties  would  now  have  been  enumerated  and  the  formal 
problem  of  constructing  the  definition  of  any  one  would  have 


4  Letters  on  Logic  to  a 

been  solved.  But,  if  a  fourth  relation,  say  that  of  being  "sub- 
sequent to"  (in  point  of  time),  were  among  those  of  the  set  to  be 
distinguished  inter  se,  our  two  principles  would  not  suffice,  for 
"subsequent  to"  is  transitive  of  any  three  events  whatever,  but 
is  not  reciprocal  of  any  two,  and  hence  "subsequent  to"  and 
"implication"  would  be  identical  in  the  sense,  that  what  is 
different  in  them  is  not  revealed  by  the  axioms  of  transitivity 
and  reciprocity  alone.  To  discover  a  character  of  difference 
between  them  it  would  be  necessary  to  examine  their  behavior 
in  the  context  of  some  third  principle. 

In  the  light  of  this  illustration  our  meaning  will  be  clear, 
when  we  say  that  a  relationship  is  defined  when  enough  of  its 
properties  have  been  enumerated  to  completely  distinguish  it 
from  whatever  other  relationships  are  in  question  and  that  the 
formal  problem  of  logic, — i.e.,  its  problem  divorced  from  ques- 
tions of  application — is  to  define  the  relationships,  of  which  it 
treats,  by  constructing  all  the  true  and  all  the  false  propositions 
into  which  these  relationships  enter  exclusively. 

The  forms  of  relationship  recognized  in  logic — we  shall  enumer- 
ate them  at  the  outset;  their  full  meaning  will  appear  in  the 
sequel — are  as  follows: 

(i)  the  categorical,  {adjective  of  quantity  and  copula), 

A(ab)  =  All  a  is  b  (is  true), 
K(ab)  =  No  a  is  b  (is  true), 
\(ab)  =  Some  a  is  b  (is  true), 
0(ab)  =  Not  all  a  is  b  (is  true), 

=  Some  a  is  not  b  (is  true), 
A'(ab)  =  All  a  is  b  (is  false), 
E'(ab)  =  No  a  is  b  (is  false), 
l'(ab)  =  Some  a  is  b  (is  false), 
0'(ab)  =  Not  all  a  is  b  (is  false), 

=  Some  a  is  not  b  (is  false) , 

(2)  the  hypothetical,  (if,  then), 

x(ab)  (is  true)  implies  y(ab)  (is  true)  is  true, 
=  If  x(ab)  (is  true)    then    y(ab)  (is  true)  is  true, 

=  x(ab)  Z  y(ab), 
x(ab)  (is  true)  implies  y(ab)  (is  true)  is  false, 

=  {x(ab)Jy(ab)V, 


YOUNG   Max   WITHOUT   A    MAST!  R 


|     the  conjunctive,  (an<h. 

x(ab)  (is  true)  and  y(ab)  (is  tru* 
=  x(ab)-y(ab), 

(4)   the  disjunctive,  (either,  or), 

Either  X  ab)  (is  true)  or  y(ab)  (is  true), 
=  x(ab)  +  y(ab). 

The  notation  x(ab),  y(ab),  etc.,  is  used  to  denote  indiffer- 
ently any  one  of  the  propositions,  A(ab),  E(ab),  I(ab),  0(ab), 
i.e.,  we  say,  x,  y.  etc.,  may  take  on  any  one  of  the  four  values 
A,E,I,0. 

In  the  categorical  proposition  x(ab)  the  terms  are  the  subject  a, 
which  is  written  first  in  the  bracket,  and  the  predicate  b,  which  is 
written  second,  i.e.,  the  term-order  in  x(ab)  is  the  order  subject- 
predicate.  When  we  wish  to  indicate  that  the  term-order  is 
not  settled,  we  shall  place  a  comma  in  the  bracket  between  the 
term-,  am/,  6)  standing  either  for  x(ab)  or  x(ba).  The  terms  a 
and  b,  stand  for  classes,  i.e.,  each  stands  for  a  group  of  ob- 
jects, which  are  conceived  by  the  aid  of  a  common  property, 
every  substantive  in  the  language  being  the  symbol  for  such  a 
group. 

A(ab)  asserts  that  all  of  the  members  of  the  a-class  are  con- 
tained among  the  members  of  the  6-class,  leaving  it  undeter- 
mined, whether  the  members  of  the  subject-class  are  related  to 
the  members  of  the  predicate-class  through  identity  or  exhaust 
only  a  part  of  the  members  of  that  class.  The  meaning  of  the 
assertion,  A(ab)  may  be  illustrated  by  the  following  diagram 
I  ig.  1  I,  i.e.,  if  All  a  is  b  is  a  true  proposition,  then  the  class  a 
is  related  to  the  class  b  in  one  of  these  two  ways. 

All  a  is  b. 


Either 


or 


The  diagrammatic  representation  of  the  other  categorical  proposi- 
tions is  given  (in  Figs.  2,  3,  4)  below. 


Some  a  is  b. 


Letters  on  Logic  to  a 
Either  la  5  J 


or      a 


b       or 


No  a  is  b. 


Some  a  is  not  b. 


Either      a 


Fig.  2. 


Fig.  3. 


or 


or 


Mor 


Fig.  4. 

It  will  be  intuitively  clear,  that  any  two  classes  whatsoever. 
a  and  b,  must  be  related  in  one,  and  cannot  be  related  in  more 
than  one,  of  the  following  five  ways  (see  Fig.  5) : 


Fig.  5. 

If  we  assert  A'(ab)  to  be  a  true  proposition,  {All  a  is  b  is  false), 
then  a  and  b  cannot  be  related  in  the  first  or  second  fashion  and 
must,  consequently,  be  related  in  one  of  the  other  three  ways 


Young  Max  without  \  Master  7 

(Fig.  5).  Glancing  back  at  Fig.  4  you  will  sit  at  once  that  the 
denial  of  A(ab),  i.e.,  the  assertion  of  the  falsity  of  A(ab),  is 
equivalent  to  asserting  the  truth  of  0(ab).  You  should  have 
no  difficulty  in  grasping  the  truth  of  the  following  equations, 
each  one  of  which  should  be  intuitively  verified  (Figs.  1-5): 

\  db)  =  0'(«.  0(ab)  =  A'(ab), 

E  <ib)  =    l'(ab),  I  ab)  =  E'(ab). 

[f  it  be  laid  down  as  axiomatic,  that  A  proposition  must  be 
either  true  or  false  and  eon  not  be  both  true  and  false,  then  a  reference 
to  the  equalities,  that  have  just  been  written  down,  will  make  it 
clear  that  A(a6)  and  0(ab)  cannot  both  be  true  and  cannot  both 
be  false  and  that  the  same  applies  to  E(ab)  and  l(ab).  Such 
propositions  as  satisfy  this  condition  are  said  to  be  contradictory. 
( Consequently  A(ab)  is  the  contradictory  of  0(ab)  and  conversely, 
while  E(ab)  is  the  contradictory  of  l(ab)  and  conversely.  In 
order  to  frame  an  image  of  this  truth  let  me  ask  you  to  glance 
again  at  Fig.  5.  You  will  observe,  since  A(ab)  and  0(ab)  have 
no  diagrammatic  representation  in  common,  that  they  cannot  be 
true  together.  At  the  same  time  it  will  be  apparent,  that,  since 
taken  together  they  exhaust  all  the  modes  of  representation  that 
there  arc  they  cannot  both  be  false.  You  will  have  no  difficulty 
referring  to  Fig.  5  again)  in  becoming  aware  that  exactly  the 
same  statements  hold  of  E(ab)  and  \(ab). 

These  result-  are  so  important  for  our  subsequent  theory, 
that  it  will  be  convenient  to  summarize  them.  In  verifying  the 
forms,  into  which  they  may  be  cast,  you  cannot  do  better  than 
refer  continually  to  the  diagrams  of  Fig.  5. 


1 


A(ab)  (is  true)   and  0(ab)  (is  true)    \>  a  false  proposition, 
is  false)  and  Q(ab)  (is  false)  Is  a  false  proposition, 

E(ab)  (is  true)    and    l(ab)  (is  true)    is  a  false  proposition. 
E(ab)  (is  false)  and    \(ab)  (is  false)  is  a  false  proposition. 


A(ab)  (is  true)    or  0(ab)  (is  true)  is  a  true  proposition, 

At  ah)  tis  false)  or  O(ab)  (is  false)  is  a  true  proposition, 

E(ab)  (is  true)  or  l(ab)  (is  true  is  a  true  proposition, 

E(ab)  (is  false)  or   I  (ah)  (is  false)  is  a  true  proposition. 

If  we  invent,  as  is  customary,  a  symbol,  (0),  to  stand  for  a 
proposition,   thai   is  false  for  all  meanings  of  the  terms,  and 


8  Letters  on  Logic  to  a 

another  symbol,  (i)  to  stand  for  a  proposition,  that  is  true  for 
all  meanings  of  the  terms,  and  if  we  recall  the  symbols,  that  have 
already  been  introduced  to  stand  for  the  conjunctive  and  the 
disjunctive  relationships,  then  the  propositions,  that  have  just 
been  enumerated,  may  be  expressed  very  concisely  thus: 


I 
II 


A  (ab)  •  O  (ab)  =  o,  E  (ab)  •  I  (ab)  =  o, 

A'(ab)  •  0'(ab)  =  o,  E'(ab)  •  V(ab)  =  o, 

A  (ab)  +  O  (ab)  =  i,  E  (ab)  +  I  (ab)  =  i, 

A'(ab)  +  0'(ab)  =  i,  E'(ab)  +  V(ab)  =  i. 


The  propositions,  which  have  just  been  written  down,  are 
fundamental  in  the  classical  logic,  the  science,  which  has  de- 
scended to  us  from  Aristotle.  The  peculiar  simplicity  of  the 
common  logic  depends  upon  the  fact,  that,  corresponding  to 
any  member  of  the  set  of  categorical  forms  there  exists  a  single 
other  member  of  the  set,  which  stands  for  its  contradictory.  Or 
(what  you  will  later  come  to  recognize  as  the  same  thing) : 
any  categorical  form,  x(ab)  (is  false),  may  always  in  this  system 
of  inference  be  replaced  by  another  categorical  form,  y(ab)  (is 
true)  (see  the  first  set  of  identities,  p.  7).  There  is  small  doubt 
that  this  was  the  motive,  which  guided  Aristotle  to  select  his 
four  forms  in  this  particular  way.  It  was  not  because  each  one 
happened  to  have  a  convenient  verbal  expression,  as  some  not 
too  discerning  depreciators  of  the  ancient  scheme  of  inference 
have  said. 

About  the  middle  of  the  last  century  a  famous  logician,  Sir 
William  Hamilton,  proposed  to  replace  the  four  forms  of  Aristotle 
by  a  new  set  of  eight.  All  but  four  of  these  eight  are  unnecessary, 
but  we  can  easily  show  that  the  four  that  are  essential  to  the 
Hamiltonian  system  are  logically  equivalent  to  the  four  tradi- 
tional ones, — i.e.,  each  member  of  the  new  set  can  be  represented 
in  the  members  of  the  old  set  and  conversely.  In  the  proposi- 
tions, A,  E,  I,  O,  the  word  some,  which  is  explicitly  stated  before 
the  subject  of  I  and  O  and  which  is  understood  but  not  expressed 
before  the  predicate  of  A  and  I,  means  some  at  least,  possibly  all. 
Should  we  understand  the  word  some  to  mean  some  at  least, 
not  all,  the  manner  in  which  the  subject  a  is  related  to  the  predi- 
cate b  in  each  one  of  the  diagrams  of  Fig.  5,  can  be  expressed  in  a 
simple  verbal  phrase.     Thus, 


Y0!  SC    Man    \\  [THOU1     \    MASTER 


All  a  is  all  l>, 

=  a(ab) 


Fig.  6. 


Some  a  is  some  b, 
=  fiiab) 


Fig.  7. 


All  a  is  some  b, 
=  y(ab) 


Fig.  8. 


No  a  is  b, 
=  *(ab) 


Fig.  9. 


All  b  is  some  a, 
=  y(ba) 


I  [G.    IO. 


These  four  forms,  which  we  have  represented  by  a,  /3,  7  and  «, 
are  the  ones,  which  Hamilton  insisted  should  be  substituted  for 
the  traditional  ones,  A,  E,  1,0.  VmU  should  have  little  difficulty, 
in  tin-  light  «.t"  what  has  gone  before,  in  understanding  the  follow- 
ing equalities,  which  represent  each  member  of  the  new  set  in 
the  members  of  the  Aristotelian  set  and  conversely: 


io  Letters  on  Logic  to  a 

A(ab)  =  a(ab)  -f  y(ab)  (see  Figs,  i,  6,  8), 

E(ab)  =   e(ab)  (see  Figs.  3,  9), 

l(ab)  =  a(ab)  +  P(ab)  +  y(ab)  +  y(ba)  (see  Figs.  2,  6,  7,  8,  10), 

0(a&)  =  e(ab)   +  j3(a&)  +  y(ba)  (see  Figs.  4,  7,  9,  10), 

a(ab)  =  A(ab)-A(ba), 
P(ab)  =    l(ab)-0(ab)-0(ba), 
y(ab)  =  A(ab)-0(ba), 
e(ab)   =  E(ab). 

While  these  tvvo  sets  of  equalities  express  a  certain  equivalence 
between  the  new  and  the  old  ways  of  formulating  the  same 
system  of  inference,  each  set  would  be  found  to  possess  certain 
advantages  peculiar  to  itself. 

You  have  now  seen  that  the  categorical  forms,  A,  E,  I,  O, 
are  composed  of  the  terms,  (a  and  b),  an  adjective  of  quantity 
{all,  some,  not  all),  and  the  copula  (is),  and  that  the  word  some 
is  always  to  be  interpreted  to  mean  some  at  least,  possibly  all 
(this  meaning  of  the  word  being  unambiguously  forced  on  us  by 
the  propositions,  which  we  say  shall  be  true  or  untrue  in  our 
science).  In  our  next  letter  we  shall  begin  to  acquaint  you 
with  some  of  the  simpler  types  of  inference^ — i.e.,  with  those 
types  of  proposition,  into  which  the  hypothetical  relationship, 
(if,  then)  implies,  enters. 


Young  Man  without  a  Master  u 


III 

In  the  hypothetical  proposition,  x  implies  y,  i  It  x  is  true  then 
y  is  true),  or,  in  our  abbreviated  notation,  x  z  y,  the  part,  !.v), 
to  the  left  of  the  implication  sign,  (z),  is  called  the  antecedent 
and  the  part,  (y),  to  the  right  of  the  implication  sign,  (z),  is 
called  the  consequent. 

Here  x  or  y  nun-  represent  any  sort  of  proposition,  but,  if  each 
one  happens  to  stand  for  a  single  categorical  form,  then  we  should 
replace  x  Z  y  by  the  more  definite  notation,  x(a,  b)  Z  y(a,  b). 
Any  implication  of  this  specific  type  is  known  as  immediate 
inference. 

You  will  recall  that  the  comma  in  the  bracket  between  the 
terms  is  used  in  order  to  indicate  that  the  term-order  is  not  settled. 
The  proposition,  x(a,  b)  Z  y(a,  b),  may  have  either  one  of  two 
fornix  If  the  term-order  in  the  antecedent  is  the  same  as  the 
term-order  in  the  consequent,  i.e.,  if  .v(a,  b)  z  y(a,  b)  be  written 

either  (a)     x(ab)  Z  y(ab), 
or         (b)     x(ba)  Z  y(ba), 

then  x  a,  b)  Z  y(a,  b)  is  said  to  be  expressed  in  the  first  figure  of 
immediate  inference.  If  the  term-order  in  the  antecedent  is  the 
reverse  of  the  term-order  in  the  consequent,  i.e., if  x(a,b)  z  y(a,  b) 
be  written 

either  (c)     x(ab)  Z  y(ba), 

or  d)     x(ba)  z  y(ab), 

then  x(a,  b)  z  y(a,  b)  is  said  to  be  expressed  in  the  second  figure 
of  immediate  inference. 

Just  as  the  comma  between  the  terms  means  that  the  term- 
order  is  not  settled,  so  the  x  in  x(ab)  and  the  y  in  y(ab)  is  used  to- 
indicate  that  the  single  categorical  form,  for  which  x(ab)  or  y(ab)' 
stands,  is  not  specified.  By  giving  specific  values  to  x  and  y 
we  shall  obtain  sixteen  distinct  propositions  and  these  sixteen 

will  l>r  all  that  exist  of  the  form,  %{at  1>)   Z  y(a,  b),  i.e.. 

A(a,  b)  z  A(a,  b)  E(a,  b)  z  A(a,  b) 

A  a,  M  z  £(o,  6)  E(a,  b)  z  E(a,  6) 

A(a,  />)  z  I(a,&)  E(a,  6)  z    I  (a,  ft) 

A(a.  b)  z  0(a,  6)  E(a,  6)  z  0(a,  6) 


12  Letters  on  Logic  to  a 

I  (a,  ft)  Z  A(a,  ft)  0(a,  6)  Z  A(a,  b) 

l(a,b)  z  E(a,  6)  0(a,  6)  Z  E(a,  6) 

I  (a,  6)  Z    I  (a,  6)  0(a,  ft)  Z    I  (a,  ft) 

I  (a,  ft)  Z  0(a,  ft)  0(a,  ft)  Z  0(a,  ft) 

These  are  obtained  by  taking  all  the  permutations  of  the  four 
letters,  A,  E,  I,  O,  two  at  a  time  and  by  taking  each  letter  once 
with  itself.  It  will  be  convenient  from  time  to  time  to  leave 
unexpressed  the  implication  sign,  (z),  and  the  part,  (a,  ft), 
and  to  write  down  the  same  set  of  sixteen  implications  in  the 
following  more  abbreviated  fashion. 


AA 

EA 

IA 

OA 

AE 

EE 

IE 

OE 

AI 

EI 

II 

OI 

AO 

EO 

10 

00 

Each  proposition  of  the  set  may  be  expressed  in  either  the 
first  or  in  the  second  figure  and  there  are,  consequently,  thirty- 
two  possible  propositions,  x(a,  ft)  Z  y(a,  ft).  The  entire  set  of 
thirty-two  is  said  to  constitute  the  array  of  immediate  inference. 
Each  member  of  the  array  is  called  a  mood  of  the  array.  The 
true  propositions  of  the  array  are  called  valid  moods  of  the  array. 
The  remaining  moods  are  called  invalid  moods  of  the  array. 

I  shall  now  ask  you  to  construct  the  array  for  yourself  and  to 
pick  out  by  inspection  the  valid  moods  in  both  the  first  and 
second  figures.     For  convenience  of  reference  Fig.  5  is  again 


OO 


Fig.  5. 


placed  on  the  page  before  you  and  it  is  to  this  that  you  must 
continually  direct  your  attention.  You  will  discover  that  the 
array  contains  ten  valid  and  twenty-two  invalid  moods.  Three 
illustrations  of  how  to  apply  the  method  of  inspection  will  furnish 
you  with  a  clue  to  the  whole  exercise,  which  I  propose. 

(1)  Consider  the  mood,  E(aft)  z  O(aft),  or,  in  our  abbreviated 
notation,  EO  in  the  first  figure.  This  asserts  that  if  a  is  related 
to  ft  as  in  the  fifth  diagram  (Fig.  5),  then  a  is  related  to  ft  either 


Vbl  M.    Man   w  iiikh  i    a   MASTl  R 


13 


a-  in  the  third  diagram  (Fig.  5)  or  as  in  the  fourth  diagram 
(Fig.  5)  or  as  in  the  fifth  diagram  (Fig.  5).  It  is  intuitively 
evident,  then,  that  EO  in  the  first  figure  is  a  valid  mood. 

Consider  the  mood,  l(ab)  z  l(ba),  i.e.,  II  in  the  second 
figure. 

I    ib  I  is  represented  by 


Either 


or 


or 


and  l(ba)  is  represented  by 


Either 


or 


or 


or 


These  two  modes  of  representation  are  identical,  except  that 
the  diagrams  do  not  appear  in  the  same  order.  But,  since  the 
order,  in  which  the  diagrams  appear,  is  irrevelant,  it  is  intuitively 
clear  that  the  mood  is  valid.  It'  we  had  chosen  to  consider  the 
mood,  \\  in  the  second  figure,  it  would  have  appeared  at  once 
that  the  diagrammatic  representation  of  the  antecedent  would 
not  have  been  the  same  as  that  of  the  consequent  and  that 
the  mood  is  invalid. 

In  general,  it'  the  m<  ab)  Z  x(ba)  is  valid,  then  x(ab)  is 

said  to  be  a  convertible  form.     The  operation  of  simple  conversion 


14  Letters  on  Logic  to  a 

consists  in  the  interchange  of  subject  and  predicate.  You  will 
discover  that  this  operation  is  permissible  in  the  case  of  E(ab) 
and  l(ab)  but  not  in  the  case  of  A(ab)  or  0(ab).  Employing 
this  language,  it  is  customary  to  say  that  E(ab)  and  l(ab)  are 
convertible  forms  or  that  K(ab)  and  l(ab)  are  simply  convertible. 

(3)  Consider  the  mood  0(ab)  Z  E(6a),  or  OE  in  the  second 
figure.  If  a  and  b  are  related  as  in  the  third  diagram  (Fig.  5) 
or  as  in  the  fourth  diagram  (Fig.  5),  then  0(ab)  is  true  and  E(ba) 
is  false.     Consequently  the  mood  is  invalid. 

This  case  leads  us  to  make  an  important  observation.  Since 
true  means  necessarily  true  and  untrue  means  not  necessarily  true, 
it  is  enough  to  point  out  one  diagrammatic  representation  of  the 
antecedent,  which  at  the  same  time  is  not  a  diagrammatic  repre- 
sentation of  the  consequent,  in  order  to  become  aware  that  the 
mood  is  invalid. 

It  is  said  of  certain  treatises  of  the  Hindoos  on  geometry, 
that  the  master,  instead  of  offering  a  proof  of  the  separate 
theorems,  was  content,  after  stating  the  proposition,  to  draw  the 
figure  and  write  under  it  the  word,  "Ecce."  The  pupil  was  thus 
expected  to  gather  intuitively  the  abstract  or  general  truth  from 
the  observation  of  a  single  illustration.  You  are  well  aware  that 
the  ideal  of  the  Greek  geometers  was  to  deduce  the  theorems  of 
the  science  from  the  fewest  possible  number  of  initial  assumptions. 
Whether  this  ideal  be  a  mistaken  one  or  not,  it  has  at  least 
inspired  the  procedure  of  all  science  to  the  present  day.  For  two 
thousand  years  the  mathematical  genius  of  the  race  was  spent 
in  the  effort  to  show  that  the  truth  of  the  fifth  postulate  of 
Euclid  could  be  made  to  depend  on  that  of  the  other  four,  and 
the  proof  of  the  independence  of  this  fifth  postulate  emancipated 
mathematical  speculation  from  many  of  the  misapprehensions, 
which  had  previously  stood  in  the  path  of  its  progress. 

If  we  were  to  apply  this  historical  contrast  of  the  Greek  and 
the  Hindoo  geometers  to  ourselves,  we  might  say  that  up  to  now 
our  study  of  logic  has  been  carried  out  on  the  Indian  plan.  Up 
to  now  we  have  been  Hindoo  logicians,  for  we  have  been  content 
merely  to  write  "Ecce"  beneath  the  diagrams  of  Fig.  5 — a  sort 
of  Cartesian  test,  an  application  of  the  dare  et  distincte  percipio. 
But  from  this  moment  forth  we  shall  fashion  our  doctrine  after 
the  Helladian  model.  We  shall  deduce  all  the  true  and  all  the 
untrue  variants  of  immediate  inference  by  the  aid  of  certain 
principles  from  the  fewest  possible  number  of  initial  postulates. 


Young  Man  withoui  a  Master  15 

The  meaning  of  the  symbol  (0)  has  been  explained  already. 
You  will  do  well,  however,  in  interpreting  the  postulates,  which 
follow,  to  translate  it  by  the  words,  an  impossibility  is  true. 
The  first  implication  below  will  read,  accordingly,  if  A{ab)  is 
true  and  0(ab)  is  true,  then  an  impossibility  is  true. 

Postulate  i.—A(ab)'0(ab)  z  0, 

Postulate  2.— A' \ab)-0\ab)  Z  0, 
Postulate  3. — E(ab)-l(ab)  z  0, 
Postulate  4  — E'(ab)-V(ab)  Z  0. 

Definition. — Two  propositions,  which  cannot  both  be  true  and 
cannot  both  be  false,  are  said  to  be  contradictory.  By  postulates 
1-4  it  follows  that  A(ab)  and  0(ab)  and  that  E(ab)  and  l(ab) 
are  contradictory  pairs. 

Principle  i. — If  in  any  valid  mood  the  antecedent  and  the 
consequent  be  interchanged  and  each  be  replaced  by  its  contra- 
dictory, a  valid  mood  will  result. 

Postulate  5. — A(ab)  z  A(ab)  is  a  valid  mood, 
Postulate  6. — A(ab)  Z  I(a^)  is  a  valid  mood, 
Postulate  7. —  l(ab)  Z    l{ba)  is  a  valid  mood. 

Theorem  1. — O(ab)  Z  0(ab)  is  a  valid  mood  (from  postulate  5 
upon  application  of  principle  i), 

Theorem  2. — E(ab)  Z  0(ab)  is  a  valid  mood  (from  postulate  6 
and  principle  i  . 

Theorem  3. — E(ba)  Z  E(ab)  is  a  valid  mood  (from  postulate  7 
and  principle  i). 

Definition. — If  x  Z  }'  is  a  valid  implication,  then  x  is  said  to 
be  a  strengthened  form  of  y  and  y  is  said  to  be  a  weakened  form 
of  .v.  By  postulate  6,  A(ab)  is.a  strengthened  form  of  l(ab)  ami 
l(ab)  is  a  weakened  form  of  A(ab);  by  postulate  7,  l(ab)  is  a 
Strengthened  form  of  I (ba)  and  \{ba)  is  a  weakened  form  of  l(ab); 
by  theorem  2,  E(ab)  is  a  strengthened  form  of  0{ab)  and  O(ab) 
i-  a  weakened  form  of  E(ab);  and  by  theorem  3,  E(ba)  is  a 
strengthened  form  of  E(ab)  and  E(ab)  is  a  weakened  form  of 
I    to). 

Principle  ii. —  If  in  any  valid  mood  an  antecedent  be 
strengthened  or  a  consequent  be  weakened,  a  valid  mood  will 
result. 

Wem   4. — A(ab)   Z   l(ba)    \-   a    valid    mood    (by    weakening 

the  consequent  of  postultae  6). 


1 6  Letters  ox  Logic  to  a 

Theorem  5. — E(ba)  Z  0(ab)  is  a  valid  mood  (by  strengthening 
the  antecedent  of  theorem  2  or  by  contradicting  and  inter- 
changing antecedent  and  consequent  in  theorem  4). 

Theorem  6. — l(ab)  Z  l(ab)  is  a  valid  mood  (by  strengthening 
the  antecedent  or  by  weakening  the  consequent  in  postulate  7). 

Theorem  7. — R(ab)  Z  E(a6)  is  a  valid  mood  (by  strengthening 
the  antecedent  or  by  weakening  the  consequent  in  theorem  3  or 
by  contradicting  and  interchanging  antecedent  and  consequent 
in  theorem  6). 

We  have,  accordingly,  by  postulating  the  validity  of  three  of 
the  moods  of  immediate  inference,  deduced  the  remaining  seven 
by  the  aid  of  two  principles.  The  deduction  of  the  invalid 
moods  I  shall  leave  to  you  as  an  exercise.  Since  it  will  be  neces- 
sary to  postulate  four  of  these  moods  as  invalid,  you  will  have 
eighteen  theorems  to  deduce.  The  postulates  and  the  principles 
of  deduction  are  given  below.  It  is  only  necessary  to  add  that 
the  additional  results  of  theorems  4-7  (above)  must  be  kept  in 
mind,  when  you  come  to  apply  principle  iv  (below). 

Postulate  8. — A(ab)  z  A(ba)  is  an  invalid  mood, 
Postulate  9. — A(ab)  Z  O(ba)  is  an  invalid  mood, 
Postulate  10. — A(ab)  Z  0(ab)  is  an  invalid  mood, 
Postulate  11. — E(ab)  Z    l(ab)  is  an  invalid  mood. 

Principle  iii. — If  in  any  invalid  mood  the  antecedent  and  the 
consequent  be  interchanged  and  each  be  replaced  by  its  contra- 
dictory, an  invalid  mood  will  result. 

Principle  iv. — If  in  any  invalid  mood  an  antecedent  be 
weakened  or  a  consequent  be  strengthened,  an  invalid  mood  will 
result. 

Theorems. — The  other  (18)  invalid  moods. 

I  shall  conclude  this  letter  by  introducing  you  to  a  form  of 
implication,  which  is  closely  allied  to  immediate  inference,  and 
I  shall  ask  you  to  pick  out  by  inspection  (by  a  reference  again  to 
the  diagrams  of  Fig.  5)  the  valid  and  the  invalid  moods. 

Already  (postulates  1-4)  we  have  had  to  interpret  specific 
instances  of  the  hypothetical  proposition,  x(a,  b)y(a,  b)  Z  0.  In 
constructing  its  array  you  have  only  to  notice  that  the  order,  in 
which  the  two  categorical  forms  conjoined  in  the  antecedent 
occur,  is  indifferent,  which  was  not  true  in  the  case  of 
x(a,  b)  Z  y(a,  b);  i.e.,  if  x  (is  true)  and  y  (is  true),  then  y  (is  true) 


Young  Man  without  a  Mash  r  17 

and  x  (is  true) — it"  two  propositions  are  represented  as  true  to- 
gether, then  this  representation  may  be  expressed  with  the 
propositions  in  either  order.  Employing  a  more  technical  lan- 
guage, we  should  sny  that  logical  multiplication  is  commutative. 
There  will,  accordingly,  be  fewer  distinct  moods  in  the  array, 
.',  b)  z  o,  than  in  the  array  of  immediate  inference. 
In  order  to  establish  the  intuitive  validity  of  any  mood  it  will 
be  enough  to  understand  that  x(a,  b)  and  y(a,  b)  cannot  be  repre- 
sented  in  any  diagram  as  true  together.  The  intuitive  invalidity 
of  any  mood  will  appear,  when  at  least  one  diagram  represents 
.vw/,  />)  and  y(a,  b)  as  true  together,  i.e.,  if  X  {is  true)  and  y 
{is  true)  docs  not  imply  an  impossibility,  then  the  mood  is  invalid. 
Clearly  a  valid  mood  will  result,  whenever  the  representations 
of  each  one  of  the  two  forms  conjoined  in  the  antecedent  do  not 
rlap  in  Fig.  5,  as  in  the  case  of  A(ab)K(ab)  Z  0.  Otherwise 
au  invalid  mood  will  result,  as  in  the  case  of  l(ab)0(ab)  Z  o. 
When  you  helve  finished  the  exercise,  which  I  have  proposed  to 
you,  you  will  have  found  that  the  array  contains  twenty  distinct 
moods,  of  which  five  are  valid  and  fifteen  invalid.  At  another 
time  we  shall  show  how  the  moods  of  this  array  may  be  deduced 
by  the  aid  of  a  principle,  which  will  be  introduced  later  on. 


i8 


Letters  on  Logic  to  a 


IV 

We  have  now  to  study  an  array  of  a  more  general  character 
than  that  of  immediate  inference  and  I  shall  begin,  not  by  de- 
scribing it  in  abstract  terms  but  by  directing  your  attention  to  a 
few  specific  instances. 

Consider  the  proposition,  A(ba)A(cb)  Z  A{ca).  Suppose  that 
we  desire  to  represent  the  antecedent  as  a  whole.  The  diagrams 
below  will  evidently  exhaust  all  the  modes  of  expression  that  are 
possible. 


Either 


or 


a]  or 


or 


You  will  observe  that  each  one  of  the  four  ways  of  representing 
the  antecedent  is  at  the  same  time  a  way  of  representing  the  conse- 
quent. Accordingly  if  a,  b  and  c  are  related  as  in  the  antecedent, 
then  it  follows  that  a  and  c  are  related  as  in  the  consequent, 
so  that  the  proposition,  A(ba)A(cb)  /_  A(ca),  is  a  valid  implica- 
tion. 

The  rule  for  constructing  the  diagrams,  which  represent  the 
antecedent  as  a  whole,  is  this:  If  the  second  form  in  the  ante- 
cedent has  (say)  three  modes  of  representation,  then  represent 
the  first  form  completely  three  times  (on  three  separate  lines)  and 
add  to  the  first  line  the  first  way  of  representing  the  second  form 
in  the  antecedent,  to  the  second  line  the  second  way,  to  the  third 
line  the  third  way.  The  antecedent  will  then  be  completely 
represented  as  a  whole. 

For  example,  consider  A(ab)0(cb)  Z  0(ca).  Since  0(cb)  is 
represented  in  three  ways,  we  represent  A(ab)  three  times,  thus: 


Young  Man  without  v  M\  rER 


19 


Now  supply  to  the  first  line  the  first  way  of  representing 0(cb),  i.e.. 


eitlu-r 


or 


and  to  the  second  line  the  second  way  of  representing  0(cb),  i.e., 


either 


or 


and,  finally,  to  the  third  line  the  third  way  of  representing  0(cb), 


i.e., 


cither   lc 


6    or 


20 


Letters  on  Logic  to  a 


You  will  perceive  at  once  that  each  separate  manner  of  denoting 
a,  b  and  c  as  related  in  the  antecedent,  is  also  a  manner  of  denoting 
a  and  c  as  related  in  the  consequent  so  that  it  is  intuitively 
evident  as  in  the  last  illustration  that  the  implication  is  valid. 
It  will  be  necessary,  perhaps,  for  you  to  study  the  more  com- 
plicated diagram  given  above  for  some  time,  in  order  to  satisfy 
yourself  that,  together  with  the  others,  it  exhausts  all  of  the 
possibilities  that  there  are. 

Since  we  have  understood  true  to  mean  necessarily  true  and  so  un- 
true to  mean  not  necessarily  true,  in  order  to  perceive  the  invalidity 
of  any  proposition  of  the  form  under  consideration,  it  will  be 
enough  to  point  to  a  single  representation  of  the  antecedent, 
which  at  the  same  time  is  not  a  representation  of  the  consequent. 

Consider  the  proposition,  K(ba)A(bc)  Z  E(ca).  The  complete 
representation  of  the  antecedent  is: 


either  [b 


or 


But  in  the  last  diagram  we  have  two  separate  instances  of  the 
untruth  of  K(ca).     Consequently,  the  implication  is  invalid. 

The  first  example,  which  we  examined  above,  was  A(ba)A(cb) 
Z  A(ca).  Consider  now  A(ab)A(bc)  Z  A(ca)  and  be  careful  to 
notice  that  the  term-order,  which  this  form  presents,  is  not  the 
same  as  that  of  the  one  first  mentioned.  The  complete  repre- 
sentation of  the  antecedent  is 


or 


or 


Y<u  v,    Man    \vi  l  HOI   l     \    MASTER  2] 

and  you  will  observe  three  distinct  instances  among  these  dia- 
grams of  the  untruth  of  the  consequent,  so  that  the  implication 
is  invalid,  i.e.,  the  validity  of  an  implication  of  the  type  under 
consideration  depends  not  only  upon  the  particular  categorical 
forms  which  enter  into  it,  but  also  upon  the  particular  manner,  in 
which  the  terms  are  arranged.  We  shall  now  determine  all  the 
possible  ways  of  arranging  the  terms.  These  will  evidently  be 
not  more  than  eight  in  number,  viz., 


ba 

ab 

ba 

ab 

cb 

cb 

be 

be 

ca 

ca 

ca 

ca 

ba 

ab 

ba 

ab 

cb 

cb 

be 

be 

ac 

ac 

ac 

ac 

In  our  last  letter  we  spoke  of  the  conjunctive  relation  of  logic 
as  being  commutative.  The  two  categorical  forms  conjoined 
in  the  antecedent  may  therefore  be  written  in  either  order  and 
we  may,  if  we  wish,  always  write  a  specific  one  of  the  two  first. 
We  agree,  as  a  matter  of  convention,  always  to  'write  first  the  form, 
which  contains  the  predicate  of  the  consequent,  thus,  A(ba)A(cb) 
Z  A(ca),  not  A(cb)A(ba)  Z  A(ca).  To  accord  with  this  con- 
vention, the  second  line  above  will  have  to  be  rearranged  thus, 


cb 

cb 

be 

be 

ba 

ab 

ba 

ab 

ac 

ac 

ac 

ac 

We  shall  now  show  thai  the  arrangements  of  this  set  are  only  a 
repetition  of  those  in  the  first  line  above,  but  in  a  different  order, 
so  that  there  will  turnout  to  be  only  four  distinct  ways  of  arrang- 
ing the  terms. 

Suppose  that  we  were  to  draw  two  lines,  one  connecting  the 
terms  in  the  categorical  form  written  first  in  the  antecedent  and 
another  connecting  the  term,  which  does  not  appear  in  the 
consequent.  Then  the  eight  varieties  of  term-order  will  appear 
thus: 


22 


Letters  on  Logic  to  a 


a 


a 


a 


a 


a> 


It  becomes  apparent  that  the  arrangements  in  the  second    r 
are  only  a  restatement  of  those  in  the  first,  for  the  figures 


give  a  very  clear  geometrical  image  of  the  number  of  possible 
term-orders.  You  will  do  well  to  commit  to  memory  at  once 
the  four  variations  in  the  first  line,  which  we  shall  constantly 
refer  to  as  figures  I,  2,  3  and  4,  respectively.  The  four  figures 
are  easily  remembered  as  combined  in  an  isosceles  triangle 
standing  on  its  vertex  (see  below). 


We  proceed  now  to  summarize  these  results  and  to  define  a 
certain  number  of  technical  terms. 

The  syllogism  is  a  form  of  implication  belonging  to  one  of  the 
types, 

1.  x{ba)y{cb)  Z.  z(ca), 

2.  x(ab)y(cb)  Z  z(ca), 

3.  x{ba)y(bc)  Z  z(ca), 

4.  x{ab)y{bc)  /_  z(ca). 

These  differences  are  known  as  the  first,  second,  third,  and 
fourth  figures  of  the  syllogism  respectively.  The  two  forms 
conjoined  in  the  antecedent  are  called  the  premises  and  the 
consequent  is  called  the  conclusion.  The  predicate  of  the  con- 
clusion is  called  the  major  term  and  points  out  the  major  premise, 
which  by  convention  is  written  first  in  the  antecedent.     The 


VOUNG    Man    WITHOUT   a    MASTER  2$ 

subject  of  the  conclusion  is  called  the  minor  term  and  points  out 
the  minor  premise.  The  term,  which  is  common  to  the  premises 
and  which  docs  not  appear  in  the  conclusion,  is  called  the  middle 
term. 

Since  x,  v  and  z  may  take  on  any  one  of  the  four  forms,  A,  E, 
I,  O,  there  will  be  sixty-four  syllogistic  variations  of  the  form, 
x(a,  b)y(b,  c)  £  z{ca),  obtained  from  the  permutations  of  the 
four  letters  taken  three  at  a  time.  Each  one  of  these  sixty-foui 
variations  may  be  expressed  in  each  one  of  the  four  figures,  so 
that  we  shall  have  two  hundred  and  fifty-six  cases  to  consider. 
Each  one  of  these  two  hundred  and  fifty-six  cases  are  known  as 
moods  of  the  array  x(a,  b)y{h,  c)  /  z(ca).  True  propositions  of 
the  array  are  known  as  valid  moods  of  the  array.  Those  remain- 
ing are  known  as  invalid  moods  of  the  array. 

In  representing  the  array  of  the  syllogism,  it  will  prove  con- 
venient, as  in  the  case  of  immediate  inference,  to  omit  the 
symbol,  (z).  and  the  parts,  (b,  a),  (c,  b),  (ca),  and  to  exhibit 
each  mood  as  a  simple  combination  of  the  three  letters.  The  best 
method  for  you  to  employ  will  be  to  add  to  each  one  of  the  sixteen 
permutations  of  the  four  letters,  A,  E,  I,  O,  taken  two  at  a  time, 
each  one  of  the  four  letters  in  succession.  The  array  under  each 
figure  will  then  appear  thus: 


AAA 

EAA 

IAA 

OAA 

E 

E 

E 

E 

I 

I 

I 

I 

O 

O 

0 

0 

AEA 

EEA 

IEA 

OEA 

E 

E 

E 

E 

I 

I 

I 

I 

O 

0 

0 

0 

AIA 

EIA 

IIA 

OIA 

E 

E 

E 

E 

I 

I 

I 

I 

O 

O 

O 

O 

AOA 

EOA 

IOA 

OOA 

E 

E 

E 

E 

I 

I 

I 

I 

O 

0 

0 

0 

24  Letters  ox  Logic  to  a 

I  shall  now  ask  you  to  construct  the  array  and  to  examine  each 
member  of  it  in  each  one  of  the  four  figures,  in  order  to  determine 
the  validity  or  invalidity  of  the  mood  in  question,  by  the  method 
of  inspection.  I  shall  furnish  you  no  clue  as  to  which  moods 
are  valid,  except  to  remark  that  six  true  propositions  will  be 
found  under  each  one  of  the  four  figures. 

That  A(ba)A(cb)0(ca)  Z  o  is  a  true  implication  will  be  intui- 
tively clear  to  you,  as  soon  as  you  have  ascertained  by  trial  that 
the  three  forms  conjoined  in  the  antecedent  cannot  be  represented 
in  any  diagram  as  true  altogether,  i.e.,  the  product  of  AAO  in  the 
first  figure  does  imply  an  impossibility. 

If  you  were  to  construct  the  array  corresponding  to 
x(a,  b)y(b,  c)z(c,  a)  Z  o  by  taking  the  permutations  of  the  four 
letters,  A,  E,  I,  O,  three  at  a  time,  you  would  discover  that  not 
all  of  the  two  hundred  and  fifty-six  moods  so  obtained  are  distinct. 
By  way  of  illustration  consider  the  three  valid  moods, 

A(ba)A(cb)0(ca)  z  o, 
A(ab)0(cb)A(ca)  z  o, 
0(ba)A(bc)A(ca)  z  o, 

Construct  a  triangle,  with  the  term  a  at  the  end  of  the  base  to 
the  right,  the  term  c  at  the  left  and  the  term  b  at  the  vertex 
above.  Let  the  arrow  indicate  the  direction  of  "flow"  from 
subject  to  predicate,  or  the  order  subject-predicate.  Then  the 
three  moods  will  be  represented  as  in  the  figures  below. 


Now  slide  the  second  and  the  third  figures  around  so  that  O 
will  appear  on  the  base,  thus: 


You  will  notice  that  in  each  instance  the  direction  of  "flow," 
as  indicated  by  the  arrows,  is  continuous  and  in  one  direction 


y<  11  we  M  w  wi  raoi  i    i  Master 


from  the  subject  of  Oto  tin-  predicate  of  <  >.     The  formal  identity 

<>i  tin-  time  cases  will  appear  more  clearly,  if  tin-  second  and 
third  figures  !><•  taken  out  of  tin-  plain-  of  tin-  paper  and  turned 
over,  thus: 


In  this  example  we  have  considered  tin-  case  of  three  valid 
mood.-,  which  at  first  blush  appeared  to  be  distinct  but  which 

turned  out  to  be  identical.  Suppose,  l>y  way  of  further  illustra- 
tion of  this  geometric  method  of  ascertaining  sameness  and  dilTcr- 
ence,  we  compare  four  invalid  moods,  vi/..  A  \  \ 

Z.  o,  in  each  one  of  the  four  figures. 

In  order  to  -how  that  each  one  of  these  Implications  i>  invalid 
it  is  enough  to  point  out  that  the  three  forms  conjoined  in 
antecedent  can  be  represented  a-  true  together  in  each  figure, 
when  the  circles  that  stand  for  a,  b  and  c  are  the  same,  ban- 
ploying  again  the  image  of  the  triangles,  we  should  ha\ 


.1 


Here  the  second  and  the  third  figures  may  be  moved  around 
before  so  a-  to  -how  an  identity  with  the  tir>t,  i.e.,  a  one-din 

tJonal  "flow"  from  the  subject  of  one  of  the  A-  to  it-  predicate, 

but  the  fourth  cannot  by  any  moving  about  be  made  to  appi 
a-  other  than  a  clockwise  or  a  counter-clockwise  "flow."     Ac- 
cordingly, of  the  four  apparent  differences,  with  which  we  began, 

Only  two  remain. 

The  i  xercise,  whit  h  1  now  propose  to  you,  is  to  construct  the 
array,   .  ._  ».  and  to  pick  out  the  valid  moods 

by  the  method  of  inspection.  You  will  recall,  that  in  order  to 
establish  the  invalidity  of  any  mood,  it  will  be  enough  to  point 
to  a  single  diagram,  which  represents  the  three  forms  conjoined 
in  the  antecedent  as  true  together.    Afterward  you  should  : 


26  Letters  on  Logic  to  a 

the  apparent  to  the  essential  differences  by  the  aid  of  the  tri- 
angles, as  described  above.  Later  on  we  shall  show  how  all  the 
valid  and  invalid  moods  of  this  array,  as  well  as  those  of  the 
syllogism,  may  be  deduced,  as  in  the  case  of  immediate  inference, 
by  postulate  and  principle. 


Ymi  MG   Man   hi  ih«»i  i    a   M  ami  k  27 


V. 

1  take  it  for  granted  that  you  have  made  a  list  of  the  valid 
moods  of  the  syllogism,  having  applied  the  method  of  inspection 
to  the  two  hundred  and  fifty-six  possible  cases.  In  order  that 
you  may  verify  your  results,  the  six  that  arc  valid  under  each 
figure  are  placed  on  the  page  before  you.    They  are: 


I 

II 

III 

IV 

AAA 

AEE 

AAI 

AAI 

AAI 

A  EO 

All 

AEE 

All 

AOO 

EAO 

AEO 

EAE 

EAE 

EIO 

EAo 

EAO 

EAO 

IAI 

EIO 

EIO 

EIO 

OAO 

IAI 

We  shall  now,  as  in  the  case  of  immediate  inference,  l>y  postu- 
lating the  truth  of  the  smallest  possible  number  of  these  mood-, 
deduce  the  remainder  by  the  aid  of  two  principles.  The  assump- 
tions, which  we  shall  have  to  make,  are  as  follow 

Postulate  1.     A(ba)A(cb)  Z  A(ca)  i-  a  valid  mood. 
Postulate  2.     E(ab)A(cb)  z  E(a»)  i>  a  valid  mood, 

Principle  i.  If  in  any  valid  mood  either  premise  and  the  con- 
clusion be  interchanged  and  each  be  replaced  by  it-  contradic- 
tory, a  valid  mood  will  result, 

Principle  ii.  If  in  any  valid  mood  a  premise  be  strengthened 
or  the  conclusion  be  weakened,  a  valid  mood  will  result, 

Theorems. — The  remaining    22    valid  moods. 

When  you  have  carefully  Btudied  tin-  examples,  which  I  shall 
down  below,  you  should  in-  able  to  carry  out  the  entire  deduc- 
tion without  further  aid,  and  the  work  of  doing  this  should  have 
been  completed  before  reading  the  remainder  of  the  letter. 

(1)  Suppose  that  we  were  to  combine  the  firsl  postulate  and 
the  first  principle.  Interchanging  the  minor  premise  and  the 
conclusion  of  A(bo)A(cb)  z  A  ca   and  replai  h  form  by  its 

Dtradictory,  we  obtain  A  '  ■  •>  (|  You  will  no' 

that  the  major  term  is  '-.  the  minor  term  ifl  (  .  and  that  the  middle 


28  Letters  on  Logic  to  a 

term  has  become  a.     The  figure  is  now  determined  in  the  way 
already  described,  viz. 


a 


a 


i.e.,  AOO  in  the  second  figure  is  a  valid  mood. 

Similarly,  by  contradicting  and  interchanging  the  major  prem- 
ise and  the  conclusion  and  replacing  each  by  its  contradictory, 
we  should  have  obtained  the  theorem: 

OAO  in  the  third  figure  is  a  valid  mood. 
You  must  in  every  case,  before  examining  the  figure,  be  sure  that 
the  major  premise  has  been  written  first. 

(2)  AOO  in  the  second  figure  being  now  established  as  a  valid 
mood,  we  may  apply  to  it  either  one  of  the  principles  in  the  same 
sense  as  to  the  postulates.  Let  us  begin  by  writing  the  mood 
with  the  terms  ordered  as  in  the  original  convention,  i.e., 
A(ab)0(cb)  Z  0(ca),  and,  applying  principle  ii,  let  us  strengthen 
the  minor  premise  O(cb)  to  E(bc).  This  will  be  possible  by 
applying  a  result  of  immediate  inference,  which  has  already  been 
established,  viz.,  E(bc)  Z  0(cb).  Accordingly  our  third  theorem 
becomes : 

AEO  in  the  fourth  figure  is  a  valid  mood. 

(3)  Suppose  that  we  were  to  turn  back  now  to  the  first  prin- 
ciple and  apply  it  to  the  result,  which  has  just  been  obtained. 
Contradicting  major  and  conclusion  and  interchanging  in 
A(ab)E(bc)  Z  O(ca)  we  obtain  immediately  A{ca)E{bc)  Z  0(ab). 

It  is  important  you  should  not  fail  to  observe  that  the  premises 
are  no  longer  in  the  normal  order  and  that  the  normal  order 
must  be  restored,  before  the  figure  can  be  ascertained.  Failure 
to  make  this  change  might  result,  as  you  will  readily  see,  not 
only  in  a  mistake  in  the  mood  but  also  in  the  figure.  Our  theorem 
is,  accordingly: 

EAO  in  the  fourth  figijfe  is  a  valid  mood. 

Had  we  chosen  to  contradict  and  interchange  the  minor  and 
conclusion  of  AEO  in  the  fourth  figure,  we  should  have  obtained, 
in  the  same  way,  the  theorem : 

AAI  in  the  fourth  figure  is  a  valid  mood. 


\'<>i  NG   Man   n  iiik.i  i    \   M  v-i  i  r 

We  observe  in  this  connection  a  general  rule  to  this  effect: 
the  application  of  principle  i  to  any  mood  in  the  fourth  figure 
places  the  premises  out  of  the  normal  order  but  leaves  the  figure 
unchanged.  Employing  a  more  technical  language  we  should 
say,  that  the  fourth  figure  is  invariant  under  principle  i. 

Having  deduced  the  twenty-two  theorems,  you  should  now 
yourself  the  exercise  of  deriving  the  valid  moods  under  each 
figure  separately  and  you  should  strive  to  arrive  .it  <  i<  h  r»  -nil 
by  the  fewesl  possible  number  of  steps.  In  deducing  those 
under  the  fourth  figure,  it  will  economize  steps  and  SO  add  to  the 
elegance  of  your  demonstration,  if  you  keep  in  mind  the  rule, 
which  has  been  stated  in  the  last  paragraph.  The  following 
rules,  whose  correctness  you  will  do  well  to  verify  for  yourself, 
show  the  effect  on  mood  and  figure  of  contradicting  ami  inter- 
changing cither  premise  and  the  conclusion. 

Major  Premise  and  Conclusion 

i      The  first   figure  changes  to  the  third  and  conversely  and 
the  premises  remain  in  normal  order. 

(2)  The  second  figure  changes  to  the  third  with  the  normal 
order  of  the  premises  reversed. 

(3)  The  fourth  figure  remains  invariant  with  the  normal  order 
ol  the  premises  reversed. 

Minor  Premise  and  Concli  sion 
i      flu-  first  figure  changes  to  the  second  and  conversely  and 

the  premises  remain  in  normal  Older. 

The  third  figure  changes  to  the  second  with  the  normal 
order  of  the  premises  reversed. 

(3)  flu-  fourth  figure  remains  invariant  with  the  normal  order 
of  the  premises  re\ ersed. 

It  will  also  In-  found  advantageous  to  state  in  the  form  of  rules 
the  effect  of  simple  conversion  in  either  premise  or  in  the  con- 
clusion.     These  rules  are: 

1     Simple  conversion  in  the  major  premise  changes  the  first 
ire  to  the  second  and  conversely,  the  third  figure  to  the  fourth 

and  conversely, 

Simple  conversion  in  the  minor  premise  changes  the  first 

figure  to  the  third  and  conversely,  the  second  figure  to  the  fourth 

and  conversely, 


30  Letters  ox  Logic  to  a 

(3)  Simple  conversion  in  the  conclusion  changes  the  first 
figure  to  the  fourth  and  conversely  and  leaves  the  second  and 
third  figures  unchanged. 

The  symbol,  (0),  we  have  employed  to  denote  a  proposition 
that  is  false  for  all  meanings  of  the  terms.  The  definition  of 
zero,  (0),  is  given  by  the  following  implications: 

0  Z  0',  O'z  0)'. 

It  is  usual  to  employ  the  symbol,  (?'),  instead  of  0' ,  which  accord- 
ingly, stands  for  a  proposition  that  is  true  for  all  meanings  of 
tiie  terms.  With  this  substitution  the  definition  of  zero,  or,  as 
it  is  sometimes  called,  the  null-proposition,  becomes: 

0  Z  i,         (i  Z  0)'. 

The  one-proposition  (i.e.,  i)  is  so  called,  because  it  acts  like  a 
unit  multiplier  in  ordinary  algebra.  When  it  appears  as  a.  factor, 
that  is,  when  it  appears  conjoined  with  one  or  more  propositions, 
it  is  usually  not  expressed,  since  it  neither  adds  to  nor  subtracts 
from  the  information  contained  in  the  product.  Its  full  meaning 
will  appear,  when  the  use  to  which  it  may  be  put  is  once  realized. 
In  order  to  illustrate  this  use  as  well  as  to  obtain  certain  con- 
structive results,  we  shall  assume  the  following  postulates,  which 
are  to  play  a  very  important  role  in  our  subsequent  theory. 

Postulates:   i  Z  A(aa),  i  Z  l{aa). 

These  postulates  mean  nothing  more  than  that  Aiaa)  and  l(aa) 
are  to  be  regarded  as  true  propositions  for  all  meanings  of  a. 
They  may  be  interpreted  to  read :  it  is  necessarily  true  that  all  a 
is  a,  it  is  necessarily  true  that  some  a  is  a. 

We  shall  now  introduce  a  principle,  which  has  been  tacitly- 
assumed  up  to  now  and  which  we  shall  have  to  make  use  of 
subsequently. 

Principle. — A  valid  implication  will  remain  valid,  when  as 
many  terms  have  been  identified  as  we  desire. 

Two  illustrations  of  the  use,  to  which  this  new  concept  of  the 
one-proposition  may  be  put,  are  set  down  below. 

(1)  By  the  principle  above  the  valid  syllogism,  A{ba)A(ch) 
Z  A(ca),  will  remain  valid,  when  the  terms  in  the  major  premise 
have  been  identified.  Accordingly,  A(aa)A(ca)  Z  A(ca)  is  a 
valid  implication.     Now  strengthen  the  part,  A(aa),  to  i  by  the 


Yot  v,    M  \\    \\  [THOU!     \    MAST]  R  }] 

first  postulate  above,  and  we  have  .    \  A  •  a  .     Hut  th< 

as  already  explained,  may  be  omitted.  The  part  thai  remains, 
A  ca)  z  A  is  <»ur  .it"  the  postulates  under  the  valid  moods 
of  immediate  inference.  This  postulate  has,  consequently,  been 
saved,  since  its  validity  has  been  made  to  depend  upon  that  of 
another,  which  was  later  introduced. 

By  the  same  principle  above  the  valid  syllogism,  I  \ 
Z  E(ca),  will  remain  valid,  when  the  terms  in  the  minor  premise 
have  been  identified.  Accordingly,  E(ab)A(bl>)  z  E-{ba)  i-  valid 
and  this  becomes  E(ab)  Z  E(6a),  when  we  suppress  the  part, 
A(bb),  as  before.  E(ab)  z  E(ba)  yields  i(ba)  z  l(ab),  when 
antecedent  and  consequent  are  contradicted  and  interchanged, 
and  the  truth  of  this  last  result,  one  of  the  moods  postulated 
under  immediate  inference,  has,  according  I  y,  been  made  to  de- 
pi  nd  upon  that  of  one  of  the  moods  postulated  later  on. 

The  only  assumption  under  the  valid  moods  of  immediate 
inference  that  remains  is,  consequently,  .\\ab)  z  \(ab).  It 
should  be  observed  in  this  connection  that  one  of  the  postulate  5 
just  enunciated  can  be  made  to  depend  upon  the  other.  For 
A(ab)  Z  I(a&)  yields  A(aa)  Z  l(aa),  for  a  =  b.  Now  weaken 
the  consequent  of  i  Z  A(aa)  to  l(aa)  and  we  have  the  theorem 
s  Z  \{aa). 

Similarly,  if  we  assume 

Postulate:   O(aa)  z  o, 

then  there  follows  immediately  the 

Theorem:    E{aa)  Z  o, 

which  is  obtained  by  strengthening  the  antecedent  in  the  postu- 
late -E  aa)  z  O(oa)  being  obtained  from  E(ab)  z  0(ab)  through 
the  identification  of ,/  and  b. 

The  use,  to  which  E(aa)  z  o  and  ()  aa)  z  o  may  be  put.  will 
be  illustrated  by  .1  reduction  of  the  number  of  postulates  for  the 
deduction  of  the  invalid  mood-  of  immediate  inferei 

3    Suppose   that    A  ab    .    I  I         and    A  ./•'<,      1  > 
valid  implications;  thenA(aa)  ..  0  mid  be  a  valid  implii 

tion  by  the  principle,  which  permits  of  the  identification  ol  the 
term-.     If  the  antecedent  of  A(aa)  .;  « I        be  strengthened  to 
then  1  /  I  I  ult-.  .md  if  the  consequent  of  i  Z.0 

weakened    to  0,   then   i  /  o   results.    Consequently,   it    A 
Z  O(ab)  and  A  ab)  1  0         were  valid,  then  ;  1  0  would 


2,2  Letters  ox  Logic  to  a 

valid.  But  (i  Z  o)'  is  part  of  the  definition  of  o.  Accordingly, 
A(ab)  /  O(ab)  and  A(ab)  Z  0(ba)  are  invalid  and  two  of  the 
postulates  for  the  derivation  of  the  invalid  moods  of  immediate 
inference  have  been  saved. 

It  is  important,  in  this  connection,  that  you  should  become 
aware  of  the  fact,  that,  while  a  valid  implication  must  remain 
true  for  all  special  values  of  the  teims  (for  example,  when  any 
number  of  the  terms  have  been  identified),  it  is  not  true  that  an 
invalid  implication  will  remain  Invalid  for  all  special  values  of 
the  terms.  Thus,  \{ab)  Z  A(ab)  is  invalid,  but  it  becomes  valid 
for  a  —  b.  This  fact  underlies  all  of  our  theory.  You  will 
easily  grasp  its  significance,  when  I  remind  you  that  true  (valid) 
means  necessarily  true,  true  in  general,  or  true  for  all  special 
cases;  whereas,  untrue  (invalid)  means  not  necessarily  true,  not 
true  in  general,  i.e.,  there  exist  at  least  some  instances  of  the 
untruth  of  the  implication  in  question,  although  not  all  special 
cases  are  necessarily  such  instances. 

I  propose  now,  as  an  exercise,  that  you  should  examine  again 
the  invalid  moods  of  immediate  inference  in  order  to  determine 
in  how  many  cases  their  invalidity  may  be  established  by  reduc- 
tion to  the  form  i  z  o,  as  explained  above. 

The  introduction  of  the  null-proposition  and  its  contradictory 
the  one-proposition  has  now  made  it  possible  for  us  to  deduce 
the  valid  moods  of  the  two  arrays,  x(a,  b)y{a,  b)  Z  o  and 
x(a,  b)y{b,  c)z(c,  a)  Z  o.  The  examples,  which  are  added  below, 
will  furnish  you  the  clue  to  the  whole  derivation.  It  will  only 
be  necessary  to  point  out  the  reason  for  identifying  i  with  the 
contradictory  of  o  and  conversely  o  with  the  contradictory  of  i. 
This  will  appear  evident,  if  we  assume  {o')'  =  o,  which  is  a 
specific  case  of  (x')'  =  x,  an  equality,  which  is  universally  true 
in  logic.     Thus,  since  i  =  o'  by  convention,  i'  =  {o')'  =  o. 

(i)  It  has  already  been  pointed  out  that  the  one-proposition 
may  be  conjoined  to  any  other  proposition;  or  suppressed,  when 
it  appears  as  a  factor  in  any  logical  product.  Accordingly,  the 
valid  mood,  A(ab)  Z  l(ab),  of  immediate  inference  may  be 
written  in  the  form,  A(ab)-i  Z  l(ab).  Now,  conceiving  the 
part,  (i),  of  the  antecedent,  as  if  it  were  a  minor  premise,  apply 
the  rule  for  contradicting  and  interchanging.  We  obtain  at  once 
A(ab)V(ab)  z  i\  or,  what  is  the  same  thing,  A(ab)E(ab)  Z  o, 
a  valid  mood  of  the  array,  x(a,  b)y(a,  b)  Z  o.     It  is  obvious  that 


\  ■  m  NG   Man   u  i  t  i k  il  i    a   Masi  i  R 

this  result  might  have  been  obtained  by  the  same  process  from 
E(ab)  Z  0(ab). 

The  principle  of  contradiction  .m<l  interchange,  which  was 
employed  in  the  deduction  of  the  valid  moods  of  immediate 
inference  and  oi  syllogism,  may  be  expn  ssed  in  a  more  general 
form,  viz.: 

If  in  any  valid  implication  the  consequent  and  any  factor  in  the 
antecedent  be  contradicted  and  interchanged,  a  valid  implication 
will  result. 

This  principle  has  been  tacitly  assumed  in  the  preceding 
ample.     The  statement  of  it  above,  its  most  general  expression, 
is  the  one  we  shall  have  to  employ  later,  when  we  approach  the 
solution  of  the  sorites,  a  form  of  implication,  whose  antecedent 
contains,  not  two,  but  any  number  of  premises. 

(2)  In  analogy  with  the  method  of  the  last  example,  the  valid 
syllogism,    A(ba)A(cb)  z  A(co),    may    be    written    A(6a)A(< 
Z  A  Contradicting  and  interchanging  the  /-factor  and  tin- 

conclusion  by  the  principle  just  enunciated,  we  obtain  imme- 
diately, A(ba)A(cb)0(ca)  z  0. 

Simple  as  the  process  is,  you  should  now  set  yourself  the  task 
of  deducing  all  of  the  valid  moods  of  the  two  arrays,  which  we 
have  just  been  considering,  assuming  the  valid  moods  of  imme- 
diate inference  and  of  the  syllogism  as  a  point  of  departure,  and 
you  should  further  strike  out  the  repetitions  l>y  the  diagrammatic 
method  explained  in  the  fourth  letter. 


34  Letters  on  Logic  to  a 


VI 

It  remains,  in  order  to  complete  the  solution  of  the  syllogism, 
to  deduce  all  of  the  two  hundred  and  thirty-two  invalid  variants 
from  the  fewest  possible  number  of  initial  assumptions.  The 
present  letter  will  be  given  over  to  the  consideration  of  this 
problem.  We  shall  find  that  two  moods  will  have  to  be  postu- 
lated as  invalid  and  that  all  of  the  others  may  be  derived  from 
these  or  else  reduced  to  invalid  moods  of  immediate  inference. 
The  mOst  elegant  way  to  proceed  will  be  to  begin  with  a  single 
postulate  and  a  single  principle  and  to  introduce  further  assump- 
tions only  when  we  are  compelled  to  do  so.  We  introduce, 
accordingly, 

Postulate  i. — E(ba)R(cb)  /.  l(ca)  is  an  invalid  mood. 

Principle  i. — If  in  any  invalid  mood  a  premise  be  weakened  or 
the  conclusion  be  strengthened,  an  invalid  mood  will  result. 

Let  us  begin  by  weakening,  in  succession,  the  major  premise  to 
E(ab),  the  minor  premise  to  E(bc),  and  finally  each  premise  to 
E(ab)  and  E(bc)  respectively.  We  shall  then  have  established 
by  postulate  and  theorem  the  invalidity  of  EEI  in  all  four  figures. 

If,  now,  the  premises  be  weakened  and  the  conclusion  be 
strengthened  in  every  possible  way,  the  untruth  of 

EEI,         EOI,         OEI,         OOI, 
EEA,        EOA,        OEA,        OOA, 

will  have  been  established  in  each  one  of  the  four  figures.  The 
invalidity  of  thirty-one  moods  has,  accordingly,  been  made  to 
depend  on  that  of  EEI  (in  the  first  figure)  alone.  It  should  be 
noted  in  this  connection  that  the  application  of  principle  ii 
(below)  to  any  mood  in  this  set  of  thirty-two  will  yield  no  mood 
that  is  not  already  contained  in  the  set;  that  postulate  2  (below) 
will  yield  no  mood  of  the  set  by  either  principle;  and  that  no 
mood  of  the  set  can  be  established  as  invalid  by  any  of  the 
methods  that  are  given  later  on.  We  now  introduce  the  second 
postulate  and  the  second  principle. 

Postulate  2. — A(ab)A(cb)  z  l(ca)  is  an  invalid  mood. 


Y<  >i  m  i   M  w  w  1 1  ik»i  i    \   Masti  b  jj 

Principle  ii. — If  in  any  invalid  mood  either  premise  and  the 
conclusion  be  interchanged  and  each  be  replaced  by  its  contra- 
dictory, an  invalid  mood  will  result. 

The  application  of  (Ins  principle  will  offer  no  difficulty  that 
has  not  been  already  overcome  and  I  have  no  doubt  that  your 
practice  in  the  derivation  of  the  valid  moods  has  been  enough  to 
enable  you  to  dispense  with  further  illustrations  here-.  Thus 
we  should  obtain  at  once  the  theorems: 

(a)  Af»E(c,  &)  z  O(ca)  by  2,  ii, 

(b)  A(ba)E(c,  b)  Z  E(ca)  by  a,    i. 

(c)  A(ab)  \(c,b)  z    l(ca)  by  b.  ii, 

(d)  l(b,a)A(cb)  z   l(ca)  by  c,   i. 

(e)  E(b,a)A(bc)  z  E(ca)  byd,  ii. 

Other  moods,  which  follow  from  the  second  postulate  and 
whose  invalidity  you  will  easily  establish  in  all  four  figure-,  are 

EIE,         IKK.         IEO,         III. 

Of  this  set  of  twenty-six  theorems,  whose  invalidity  depends 
en  that  of  A(ab)A(cb)  Z  I(fa)>  it  cai\  be  said,  that  each  one  is 
independent  of  our  original  set  of  thirty-two  and  that  none  can 
be  reduced  by  the  methods  that  we  are  about  to  introduce. 

You  will  recall  that  since  a  valid  implication  must  remain  true, 
when  a-  many  terms  have  been  identified  as  we  desire,  it  follow- 
that  the  invalidity  of  any  implication  i-  established,  whenever 
we  can  point  to  a  special  instance  of  its  being  untrue.  The 
invalidity  of  a  mood  of  the  syllogism  would  be  proven,  accord- 
ingly, if  we  could  reduce  it  to  the  particular  case  of  an  invalid 

mood  of  immediate  inference.  The  examples,  which  I  have  set 
down  below,  will  be  enough  to  suggest  to  you  a  general  method 
of  reduction,  that  will  yield  the  mood-  not  yet  resolved. 

1     Suppose  that  \<<i.  b  (  I        z  0(r </ 1  were  valid  implications. 

Identifying  terms  in  the  major  premise  and  suppressing  the  part. 

1  :  .  i.e.,  strengthening  l(aa)  to  i),  it  would  follow  th.it 
I  I  ic)  Z  Off  a)  is  a  valid  implication.  But  0(ac)  /()<,/>  is 
an  invalid  mood  of  immediate  inference  and.  consequently, 
I     .  •'  <  >  .V)  z(l         ire  invalid  moods  of  the  syllogism. 

By  the  method  of  the  I  ample,  A  a     \  A 

will  reduce  to  A'af)  z  A  ca  .  for  b  =  c,  and  A     . ■    \  A 

for    b  =  a.     The    mood-.    A  ■■■:  0  0(ca)    and    <  '     :    \ 


36  Letters  on  Logic  to  a 

Z  O(ca),  cannot  be  reduced  by  the  method  in  question,  but  they 
may  be  derived  from  the  two  moods  just  established  by  the  aid 
of  principle  ii.     Thus, 

A(ab)A(cb)  Z  A(ca)  yields  A(ab)0(ca)  Z  0(cb),  on  inter- 
changing contradictories  of  minor  and  conclusion,  and 

A(ba)A(bc)  Z  A(ca)  yields  0(ca)A(bc)  Z  0(ba),  on  inter- 
changing contradictories  of  major  and  conclusion. 

(3)  Suppose  E(ba)A(cb)  Z  l(ca)  were  a  valid  mood  and  iden- 
tify terms  in  the  minor  premise.  The  result  is  an  invalid  mood  of 
immediate  inference.  Accordingly,  E(ba)A(cb)  Z  l(ca)  is  an 
invalid  mood  of  the  syllogism.     Now 

E(ba)A(cb)  Z  \{ca)  yields  E(ba)E(ca)  Z  O(cb),  on  interchang- 
ing contradictories  of  minor  and  conclusion. 

This  last  result,  whose  invalidity  in  the  other  figures  follows 
at  once  by  principle  i,  will  yield  invalid  moods  of  the  syllogism 
that  remain  to  be  established.  We  obtain  immediately  from 
EEO,  by  principle  i,  each  one  of  the  following  moods  in  each 
one  of  the  four  figures,  viz. 

EEE,        EOE,        OEE,        OOE, 
EEO,        EOO,        OEO,        OOO. 

The  invalid  moods  of  the  arrays,  x(a,  b)y(b,  c)  Z.  0  and 
x(a,  b)y(b,  c)z(c,  a)  Z  0,  are  gotten  at  once  from  results  already 
obtained  by  the  principle  of  interchanging  contradictories,  as 
illustrated  in  the  examples  below. 

(1)  A(ab)  Z  A(ba)  may  be  written  A(ab)-l  Z  A(ba). 
Contradicting  and  interchanging,  there  results  at  once 

A(ab)A'(ba)  Z  *',         or        A(ab)0(ba)  z  0. 

(2)  A(ab)A(bc)  Z  A{ca)  may  be  written  A(ab)A(bc)  -i  Z  A{ca). 
Consequently,  as  before,  A(ab)A(bc)0(ca)  z  0. 


Yoi'Nt;  Mw  wiiiiolt  a  Master  37 


VII 

The  type  of  implication,  which  we  arc  now  to  consider,  is  one, 
in  which  the  number  of  terms  is  greater  than  three  and,  as  in 
immediate  inference  and  syllogism,  the  number  of  premises  one 
I'  ss  than  the  number  of  terms.  Accordingly,  it  will  be  more 
convenient  to  employ  in  place  of  the  term-symbols,  a,  b,  c,  etc., 
the  ordinal  numbers,  /,  2,  3,  etc. 

The  sorites  is  an  implication  of  the  general  form: 

x(i,  2)y(2,  3)z(3,  4)  ••■  u(n  -  1,  ti)  z  w(ni), 

following  the  convention  of  writing  the  major  premise  first,  so 
that  the  term-order  in  the  conclusion  is  fixed  as  (///). 

We  shall  begin  by  illustrating  the  manner  of  constructing  a 
valid  sorites  from  a  chain  of  valid  syllogisms. 

(i)  Suppose  that  we  were  to  be  given  the  chain  of  valid 
svllogisms, 

A(2i)M32)  z  A(ji), 
A(j/)A(4J)  Z  AG*/), 
A(4i)A(54)  Z  A(5/  • 

and  were  asked  what  valid  mood  of  the  sorites  is  thereby  implied. 
It  is  clear  that  the  major  premise  of  the  last  syllogism,  being  the 
ne  as  the  conclusion  of  the  second,  may  be  strengthened  to 
A  31  A  :  .  The  immediate  resull  of  this  strengthening  is  a 
valid  mood  of  the  sorites,  \i/.. 

A  31  M43)*(S4)  Z  A(5J). 

The  major  premise  of   this  last   implication   may  in   turn   be 
strengthened  to  A  21  A  32  .  by  the  firsl  syllogism,  and  we  have 

A  21  A  32  A  \3  A  54)    ■  A  51). 

j  The  valid  mood  o!"  tin-  sorites,  which  ha-  jusl  been  built  up, 
may  in  turn  be  reduced  sui  cessively  t..  each  member  "t"  the  chain, 
upon  which  it  depends.  If  tin-  terms  in  the  fourth  premise  be 
identified,  the  sorites  becom<  - 

A  j/    \  32  A  /,■    \  //-  .    A  //  . 

or,  when  we  suppress  in  the  usual  way  the  p.irt  A  : 

A  21    \    \2  A   :  A   M 


38  Letters  on  Logic  to  a 

Similarly,  identifying  terms  in  the  last  premise  of  the  last  mood, 
we  obtain 

A(2I)A(32)   z  A(j/), 

which  is  the  first  syllogism  of  the  chain. 

The  second  syllogism  will  evidently  be  gotten  by  identifying 
terms  in  the  first  and  last  premises  and  the  third  syllogism  by 
identifying  terms  in  the  first  and  second  premises. 

(3)  Another  method  of  constructing  a  valid  mood  of  the  sorites 
from  a  chain  of  valid  syllogisms  depends  upon  an  application  of 
the  following: 

Principle. — If  in  any  valid  implication  the  same  factor  be 
conjoined  to  both  antecedent  and  consequent,  a  valid  implication 
will  result. 

Let  our  chain  of  syllogisms  be 

E(2i)A(32)  z  E(jj), 
E(3i)  1(34)  Z  0(4i), 
0(4i)M45)  Z  0(5/), 

and  suppose  that  we  conjoin  to  antecedent  and  consequent  of  the 
first  member,  E(2/)A(j2)  Z  E(jz),  the  minor  premise  of  the 
second  member,  1(34),  and  so  obtain 

E(2i)A(32)l(34)  Z  E(3i)l(34)- 

The  second  syllogism  allows  us  to  weaken  the  consequent  of  this 
result  to  0(41).     Accordingly,  we  obtain 

E(2i)A(32)l(34)  Z  0(4/). 

Now  conjoin  to  antecedent  and  consequent  of  this  sorites  the 
minor  premise  of  the  third  syllogism,  A (45),  i.e., 

E(2i)A(32)l(34)A(45)  Z  0(4i)A(45), 

and  weaken  the  consequent  of  this  implication  to  0(5/)  by  the 
last  member  of  the  chain.     Consequently, 

E(2i)A(32)l(34)A(45)  Z  0(5i) 

is  the  valid  mood  of  the  sorites,  which  was  to  be  built  up. 

(4)  Suppose,  being  given  a  valid  mood  of  the  sorites,  we  should 
be  asked  to  find  the  chain  of  syllogisms,  upon  which  it  depends. 
Let  the  mood  be 

A(i2)A{23)0{43)A{45)A{56)  z  0(6i). 


V  oi  ng  Man  \\  ithoi  i    \  Master 

The  premises  <>t  the  first  syllogism  <>i  the  chain  will  In-  tin-  Nimr 
.i->  the  first  two  premises  <>t  the  smite-  and  the  minor  of  the 
second  syllogism  will  be  the  same  as  the  third  premise  "t  the 
>oriti>,  and  so  on.  The  fragmenl  <>t  tin-  chain  so  far  ascertained 
will  be 

A(i2)A(2j)  z  — 

0(43)  Z  - 

-  M45)  Z  — 

-  A:5M  Z  - 

New  the  conclusion  of  the  first  syllogism — whose  premises  appear 
out  of  the  normal  order — which  is  evidently  A(jj),  must  be  the 
major  of  the  following  syllogism,  whose  conclusion  in  turn  is 
determined  as  0(41).  Following  oul  this  same  process,  each 
member  of  the  chain  will  be  unambiguously  determined  as, 

A  12  A  23)  .    A  /. 

A(zj)Ofo)  .    0(4i), 

<>  //  A  15  -    0(5/), 

0(5i)A(56)  Z  0(6i). 

The  invalidity  of  any  mood  of  the  sorites  will  be  established 
at  once,  if  it  can.  through  the  identification  of  term-,  be  redui 
to  an  invalid  syllogism.     The  examples,  which  arc  given  below, 
will  illustrate  all  of  the  methods,  which  it  will  be  necessary  to 

employ  later  on. 

i     To  establish  the  invalidity  of  the  sorites, 

A  21  A  23  A   13  A  $4    -    A  .,-/.. 

If  the  terms  in  the  last  premise  and  in  the  next  to  the  last  premise 
be  identified  and  the  parts,  A  //  and  A  jj  .  be  suppressed,  w< 
should  obtain  in  buci  ession, 

A  21  A  23  A  u)  z  A  //  . 
A  21  A  23)  z  A(ji). 

Now  tin-  last  result,  AAA  in  the  third  figure,  is  an  invalid  syllo- 
gism and,  consequently,  the  mood  of  the  Borites  is  invalid. 

Had  we  identified  terms  in  the  first  and  last  premise  we  should 
have  obtained,  in  the  same  way, 

A  23)A(4       .     A    : 

,in  invalid  mood  in  thi  ire.     You  will  remember  thai 


40  Letters  on  Logic  to  a 

the  reduction  of  a  mood  of  the  syllogism  to  a  valid  mood  of 
immediate  inference  proves  nothing  as  regards  either  its  validity 
or  its  invalidity.  The  same  observation  of  course  holds  true 
of  a  mood  of  the  sorites.  If  in  the  case  in  question  we  had 
identified  terms  in  the  second  and  third  premises,  there  would 
have  resulted 

A(2/)A(52)  Z  A(5/), 

a  valid  mood  of  the  first  figure,  and  the  invalidity  of  the  original 
sorites  would  not  have  been  established  by  this  process. 
(2)  To  establish  the  invalidity  of  the  sorites, 

0(i2)0(23)A(34)0(45)0(56)  Z  A(6i). 

Let  us  begin  by  identifying  terms  in  the  A-premise  and  sup- 
pressing the  part,  A(jj),  and  we  have 

0(12)0(23)0(35)0(56)  z  A(6i). 

Now  strengthen  each  one  of  the  O-premises  to  E-premises  and 
our  result  will  be 

E(i2)E(23)E(35)E(56)  z  A(6i). 

Finally,  identifying  6  and  3  and  5  and  2,  and  converting  simply 
in  the  third  premise,  the  mood  becomes 

E(i2)E(23)E(23)E(23)  z  A(3i). 

It  will  be  necessary  now  to  postulate  the  implication, 

E(ab)  z  E(ab)E(ab), 

which  we  shall  have  occasion  to  employ  continually  later  on. 
By  its  aid  we  may  strengthen  E(2j)E(2j)  to  E(2j),  so  that,  by 
two  steps,  we  arrive  at 

E(72)E(2j)  z  A(3i), 

an  invalid  mood  of  the  syllogism.  The  mood  of  the  sorites,  with 
which  we  began,  is,  accordingly,  invalid  as  well. 

In  the  light  of  these  illustrations  it  will  be  clear  that: 

(a)  The  validity  of  a  mood  of  the  sorites  containing  A-  or 
I-premises  may  be  made  to  depend  on  the  validity  of  a  mood,  in 
which  these  premises  are  absent. 

(b)  Since  an  O-premise  may  always  be  strengthened  to  an 
E-premise,  the  validity  of  a  mood  of  the  sorites  containing  E- 


Vol  NG    M.w    u  [TH0U1     \    MAS!  4] 

or  O-premises  maj  be  made  to  depend  on  the  validity  of  a  mood, 

in  which  only  one  E-premise  occurs. 
In  general,  if  the  chain  of  syllogisms, 

J,  2)  z  .v3(j/), 
x3(ji)x*{4,  3)  Z  x,{4i), 
.v5U/).vc(5,  4)  z  .y7(5j), 


x-.k-b(n  —  1  /).Von_4(//,  n  —  1)  Z  Xsn_3 (ni), 

be  valid  throughout,  then 

.v,(2,  7).v2(j,  2)34(4,  j)  •  •  •  Xtn-4(n,  n  -  1)  Z  *i*_*(ni) 
is  a  valid  mood  of  the  sorites.  Hence  a  certain  number  of  valid 
sorites  may  be  constructed  from  chains  of  valid  syllogisms. 
//  remains  to  be  proven  that  the  only  valid  moods  of  the  sorites  that 
exist  may  be  built  up  from  chains  of  valid  syllogisms  in  the  manner 
described. 

We  proceed  to  enunciate  certain  theorems,  which  will  abbre- 
viate the  work  of  establishing  the  general  solution  that  follow-. 

Definition. — A  form,  x{ab),  which  becomes  unity,  when  its 
terms  are  identified,  is  said  to  be  affirmative.  By  results  already 
obtained,  A(ab)  and  l(ab)  are  affirmative  while  E(ab)  and  0(ab) 
are  not  affirmative  forms. 

Definition. — A  form,  x(ab),  which  becomes  null,  when  its  terms 
are  identified,  is  said  to  be  negative.     By  results  already  obtained, 
K(ab)  and  0(ab)  arc  negative,  while  .\  ab)  and  l(ab)  are  not 
iti\ e  forn 

Definition. — If  x(ab)  z  }'(ab)  and  ,y(ab)  Z  x(ab)\'  then  x(ab) 
is  said  to  be  universal  and  \  ab)  is  said  to  be  particular.  Accord- 
ingly, A(ab)  and  K(ab)  are  universal,  l(ab)  and  0(ab)  are  par- 
ticular forms. 

Theorem  1. — If  the  conclusion  be  affirmative,  then  all  of  the 
premises  are  affirmative.     I  <>r,  if  there  were  present  one  or  more 

j  iti\c  premises,  the  mood  of  the  sorites  would  1"'  reducible  t<> 

an  invalid  syllogism  of  one  <»f  the  forms, 

AEA,        EAA,        AEI, 

I    \I.         I  I. A,         I  I  I 

'-If  the  conclusion  be  ■  '.  then  one  and  only  one 

premist  I  or,  it  all  of  the  premisi  -  w<  re  affirmative, 

the  mood  of  tin-  sorites  would  be  reducible  to  an  invalid  sj  llogism 
of  one  of  the  forms, 


42  Letters  on  Logic  to  a 

AAE,        AIE,        IAE,         HE, 
AAO,        AIO,        IAO,        IIO, 

and,  if  more  than  one  premise  were  negative,  to  one  of  the  forms, 

EEE,        EOE,        OEE,        OOE, 
EEO,        EOO,        OEO,        OOO. 

Theorem  3. — //  the  conclusion  be  universal,  then  all  of  the 
premises  are  universal. 

Theorem  4. — If  the  conclusion  be  particular,  not  more  than  one 
premise  is  particular. 

It  will  be  convenient  to  take  the  conclusion  successively  in 
each  one  of  the  four  forms. 

Conclusion  in  the  A-form 

The  conclusion  is  universal  and  affirmative.  Accordingly,  the 
premises  are  all  universal  (theorem  3)  and  all  affirmative  (theorem 
1).     The  sorites,  if  it  be  valid,  must,  consequently,  be  of  the  form, 


A(7,  2)A(2,  3)  ...  A(n-  j,  n)  z  A(w/). 

Now  the  term-order  in  each  premise  is  established  as  (s  s  —  1), 
i.e.,  with  the  larger  ordinal  number  coming  first,  for,  suppose 
that  the  term-order  (s  —  1  s)  should  occur  in  any  premise. 
Then  the  mood  of  the  sorites  would  be  reducible  to  an  invalid 
syllogism, 


Either  A(s  —  i,s  —  2)  A  (5  —  is)  Z  AQ  5  —  2), 
or     A(s  -  1  s)A(s  +  1,  s)  z  A (7+7  5  -  1). 

The  term-order  is,  accordingly,  no  longer  ambiguous  and  the 
sorites  is  of  the  form, 


A(2/)A(j2)A(4j)  •■■  A(nn-  1)  Z  A(m), 

which  can  be  generated  from  the  chain  of  valid  syllogisms, 

A(27)A(J2)  z  A(ji), 
A(3i)A(43)  Z  A(4i), 
A(4i)A(54)  Z  A(5i), 


A(n  -  1  i)A(n  n  -  1)  Z  A(w/). 


\.  'i  \i,   M  w   \\  i  iiioi  i    a    MAST!  R  4;> 

Coni  i.i  SION    IN    i  Hi.    l-i  I  'k\i 

The  conclusion  is  affirmative,  so  that  all  of  the  premises  are 
affirmative  {theorem  I),  and  particular,  so  that  not  more  than 

one  premise  is  particular  (theorem  4). 

Case  I—  A(/,  2)\  _\  3)  •  •  ■  An  -7,  n)  z  !(*/),  a  form  which 
contains  no  [-premise. 

The  first  premise,  which  presents  it>  terms  in  tin-  order 
(s  —  1  s),  establishes  that  order  in  each  premise,  which  follows. 
For,  suppose  a  premise  coming  after  the  premise  in  question 
would  present  the  term-order  (s  s  —  1).  Then  the  sorites  would 
be  reducible  to  au  invalid  syllogism  of  the  form, 


A(s  -  2  s  -  i)A(s  s  -  1)  Z  10  5  -  2). 


Let  us  suppose  that  the  term-order  (s  s  —  1)  is  preserved  .1-  far 
as  the  rtli  premise  and  is  then  reversed.  Then  the  order 
(s  —  1  s)  is  established  from  the  rth  premise  to  the  end  and  the 
sorites  becomes 

A  21  .V32)  ■  ■  ■  A(>  r  -  i)A(r  r  +  1)  ■  ■  ■  A(n  -  1  n)  I  i(m), 
which  is  constructible  from  the  chain, 

A  21)A(J2)  Z  A(ji), 


A(r  -  1  j)A(r  r  -  1)  Z  A(rj), 

A    i)A(r  r  +  1)  Z  I(r  +  /  /), 

I  r  +  /  z)A(r  +  /  r  +  2)  z  I(r  +  2  /  . 


I(>/  -  /  1  A  ;..  -  /  n)  z  I(ni). 


If  the  term  order  (5  s  —  1)  is  preserved  from  the  first  premise  to 
the  last,  so  that  the  form  of  the  sorites  is 


A  j/)A(j2)  •  •  •  A(«  n  -  1)  z  !(«/), 
then  the  generating  chain  of  syllogisms  will  be 

A  21  A  32)  Z  A(JJ), 

A  jl  A  tJ)  Z  AC//), 


A  >•      2  1  A  1  -  in  —  2)  £  A';/  -  /  /  . 

A  n  -  /  /  A  n  n  -  1)  £  l(ni). 

II. — Suppose  that  the  /th  premise  i^  in  the  [-form  and 

that  the  sorites  i- 


A(/,  2-  •  •  •  At  -  1.  I  I  /.  /  -f  d  •  •  •  A(«  -  j,  n)  Z  1 


44  Letters  on  Logic  to  a 


The  term-order  in  the  first  t  —  i  premises  is  established  as 
{s  s  —  i).  Otherwise,  by  the  identification  of  terms,  we  should 
come  upon  an  invalid  syllogism  of  the  form, 


A{s  -  i  s)l{s,  s  +  i)  Z  l(s  +  i  s  -  i). 


Similarly,  the  order  of  terms  in  the  last  n  —  t  —  i  premises  is 
established  as  (s  —  i  s).  For,  should  any  premise  following  the 
I -premise  present  the  term-order  (s  s  —  i),  the  mood  of  the 
sorites  would  reduce  to  an  invalid  syllogism,  viz., 


1(5  —  2,  s  —  i)A(s  s  —  1)  Z  I(s  s  —  2). 
Consequently,  the  sorites  takes  the  form, 


A(2i)  •  •  •  A{tt  -  i)\{t,  t  +  i)A{t  +  1  t  +  2)  •  •  • 

A(«  —  in)  /  \{ni), 

which  may  be  built  up  from  the  chain  of  syllogisms, 

A{2i)A(32)  z  A(jj), 


A(/-/  i)A{t  t-  1)  Z  A(/j), 
A(ti)l(t,  /+7)  Z  I(H-  1  1), 

i(T+i  i)A(i+i  TT~2)  z  i(T+2  1), 


I(«  —  1  i)A(n  —  1  n)  z  I(wj). 

Conclusion  in  the  E-form 

Here  there  must  occur  a  single  E-premise  {theorems  2,  3)  and 
all  of  the  other  premises  are  in  the  A-form  {theorem  3).  All  valid 
moods  of  this  form  are  to  be  obtained  by  contradicting  and  inter- 
changing the  I-premise  and  the  I-conclusion  in  the  type  of  valid 
sorites,  which  has  just  been  established  (case  II,  above).  For, 
every  mood  of  the  sorites  with  an  E-conclusion  not  so  obtained 
would  be  reducible  to  an  invalid  mood  already  established  by 
contradicting  and  interchanging  the  E-premise  and  the  con- 
clusion. 

Conclusion  in  the  O-form 

One  and  only  one  premise  is  a  negative  form.  All  valid  moods 
of  this  type  are  obtained  by  contradicting  and  interchanging  a 
premise  and  conclusion  in  one  or  other  of  the  valid  moods  already 


Yi  A  SC    Man    u  i  I  B01  I    A    MAS1  BR  45 

il)li>lu'(l.  For,  t>tlRT\vi>i\  .1-  in  the  last  case,  we  could  reduce 
any  mood  not  so  derived  to  one  of  the  invalid  moods  already 
established. 

Accordingly,  .ill  of  the  valid  moods  of  the  sorites  are  deter- 
mined as  of  certain  specific  types  and  each  one  <>l"  these  moods 
may  be  constructed  from  .1  chain  of  valid  syllogisms 


46  Letters  on  Logic  to  a 


PROFESSOR   SINGER'S  SYLLABUS 

(I) 
Classification  of  Sciences  into 

Empirical:   Physics,  Chemistry,  Biology,  Sociology,  etc. 
Non-Empirical: 

(a)  In  which  judgment  of  truth  of  propositions  involves  knowl- 
edge of  the  meaning  of  terms:    (Mathematics). 

(b)  In  which  judgment  of  the  truth  of  propositions  does  not 
involve  knowledge  of  the  meaning  of  terms:    (Logic). 

Definition  of  Logic:  Logic  is  the  science  whose  problem  it  is  to 
construct  all  propositions  whose  truth  is  independent  of  the 
meaning  of  terms. 

(ID 

Grammarians  recognize  six  kinds  of  sentence:  Declarative. 
Optative,  Exclamatory,  Interrogative,  Hortatory,  Imperative, 
These  fall  into  two  classes: 

(a)  Sentences  that  are  either  true  or  false.     (First  three.) 

(b)  Sentences  that  are  neither  true  nor  false.     (Last  three.) 
Definition  of  Proposition:   A  proposition  is  a  sentence  that  is 

either  true  or  false.     The  logician  recognizes  the  following  forms 

of  propositions  as  necessary  and  sufficient  for  the  expression  of 

any  truth. 

Categorical        All  a  is  b  =  A  (a  b) 

No  a  is  b  =  E  (a  b) 

Some  a  is  b  =  I  (a  b) 

Some  a  is  not  b  =  O  (a  b) 

Hypothetical     X  implies  Y  =  X  z  Y 

X  does  not  imply  Y  =  (X  z  V)' 

Conjunctive      X  (is  true)  and  Y  (is  true)  =  X- Y 

Disjunctive       Either  X  (is  true)  or  Y  (is  true)  =  X  +  Y. 

(Ill) 

Categorical  forms  composed  of  terms  and  relation. 
Terms  are  subject  (a)  and  predicate  (b). 
Relations  are  composed  of 


V  'i  \.  i   Man   Wl  ni<  n  i    \    M  \-i  I  R  (7 

Adjective  of  quantity  and        Copula 

A  all  is 

E  no  is 

I  sonii-  is 

0  not  all  is 

The  array  of  propositions  of  categorical  forms  in  which  terms 
arc  identical : 

X  (a  a) 

A  (a  a)  E  (a  a) 

1  (a  ..  0    a 

Definition:  True  propositions  "i  a  given  array  are  called  valid 
moods  of  that  array. 

False  propositions  <>f  a  given  array  are  called  invalid  moods 
of  that  array. 

Post,  i .  A  a  a    is  \  alid  mood. 

Post.  2.  I  a  a) 

Post.  3.  E(a  a)  "  invalid   mood. 

Post.  4.  0  a  a) 

(IV) 

Definition  1. — In  the  hypothetical  forms  X   /  Yan<l<X  .    Y 
X  is  called  the  antecedent  and  Y  is  called  the  consequent. 

nition  J.— If  X  Z  V  and  Y'    /  X,  X  i-  called  the  contra- 
dictory of  Y. 

Principle    i,  (X  I   Y'<   I   (Y   ^  X 
Principle  ii.  iY'  Z  X)   Z   (X'  Z  V 

Theorem:  If  X  is  the  contradictory  of  Y.  then  Y  is  the  contra- 
dictory of  X . 

P  rt.  1.    A  .1  b)  z  O'   a  b  .  Post.  2.    I  >' (a  b)  .    A    1  b). 

Post.  3.     E  a  1"  z    I'  (ab).  Post.  4.      I'(ab)  zE    1  b  . 

Historical  note:  There  are  m»  other  pairs  of  contradictories 
among         jorical  forms.     Contrary  forms,  sub-contrary  forms, 

alternate  form-. 

\ 

The  array  X    ...  b    .    Y    ...  I 
Two  figures  1.  X  .1  !>•  .    Y    ab 
2.  X  a  b)  .;  Y    ba 


48  Letters  on  Logic  to  a 

Sixteen  moods  in  each  fig. 

The  valid  moods. 

Prin.   i,  (X  Z  Y)  Z  (Y'  Z  X')  Denial  of  consequent. 

Prin.  II,  (X  Z  Y)(Y  Z  Z)  Z  (X  Z  Z).     Transitivity. 

Post,  i,  A  (a  b)  z  I  (a  b)         (A  I)i  is  valid  mood. 

Post.  2,   I(ab)  Z  I  (ba)         (I   I)2  "      " 

Deduction  from  Post.  I  and  2  and  from  Posts,  of  (IV)  of  valid 
moods  by  means  of  Prins.  i  and  ii. 

Valid  moods:     (A  A)1  (A  I)j  (E  E^  (EO)!  (I  I)i  (O  0)i 

(A  I)2  (E  E)2  (EO)2  (I  I)2 
The  Invalid  moods: 

Prin.  iii.  (X  z  Y)'  Z  (Y'  Z  X')'. 

Prin.  iv.  (X  Z  Z)'(Y  z  Z)  Z  (X  Z  Y)'. 

Prin.  v.  (X  Z  Y)(X  Z  Z)'  Z  (Y  Z  Z)'. 

Post.  i.  {A(a  b)  z  0(a  b)}'     (A  0)i  is  invalid. 

Post.  2.  |E(ab)  z    I(ab)}'     (E   I)x  " 

Post.  3.  {A(a  b)  z  A(b  a)}'     (A  A)2  " 

Post.  4.  {A(a  b)  z  0(b  a)}'     (A  0)2  " 

Rules  for  immediate  detection  of  invalid  moods: 

Def.  i.  A  form  which  yields  a  true  proposition  when  the 
terms  are  made  identical,  is  called  an  affirmative  form. 

From  (III),  A  (a  b)  and  I  (a  b)  are  affirmative  forms. 

Def.  2.  A  form  which  yields  a  false  proposition  when  terms 
are  made  identical  is  called  a  negative  form.  From  III,  E(a  b) 
and  0(a  b)  are  negative  forms. 

Rule  I.     An  affirmative  form  does  not  imply  a  negative  form. 

ex.  (A  0)2 
Rule  II.     A  negative  form  does  not  imply  an  affirmative  form. 

ex.  (E  I)i 

Def.  3.  If  X(a  b)  z  Y(a  b)  and  {Y(a  b)  z  X(a  b)}'  X(a  b) 
is  called  a  universal  form  and  Y(a  b)  is  called  a  particular  form. 

From  results  of  present  chapter  A(a  b)  and  E(a  b)  are  universal 
forms,  I  (a  b)  and  0(a  b)  are  particular  forms. 

Def.  4.  The  subject  of  a  universal  and  the  predicate  of  a 
negative  form  is  called  a  distributed  term,  other  terms  undis- 
tributed. 

Aff.  Neg. 

Universal A  (a  b)  E(a  b) 

Particular I  (a  b)  0(a  b) 


Yi  >ung  M  w  wn  hol  i    \  Masi  ik  49 

Rule  III.     A  form  In  which  a  given  term  is  undistributed  does 
not  imply  a  form  in  which  that  same  term  is  distributed.      A  A 
Proof  of  tlic  necessity  and  sufficiency  of  these  rules. 

(VI) 

The  array  X(a,  b)  Y(b,  c)  z  Z(c,  a) 

Def.   i.    When  forms  are  conjoined  in  antecedent,  each  is 

called  a  premise  and  the  consequent  is  then  called  the  conclusion. 

Def.  2.    The  order  of  terms  in  present  array  called  cyclical 

order. 

Def.  3.     A  cyclical  order  of  terms  with  one  premise  called 

immediate  inference.     X(a,  b)  z  Y(a,  1".     With  more  than  one 
premise,  mediate  inference.     Mediate  inference  with  two  premi 
called  syllogism  (.  z  syn  and  logos);   more  than  two  forms  called 
Sorites  (  Z  soros) . 

(1)  In  order  of  premis< 

X  a,b)  V  b,  c)  .    /  c,a) 
Y(b,  c)  X(a,b)  Z  /  C, 

2    In  order  of  terms. 

I.  X(ba)  V(cb)  z  Z(ca) 

II.  X(a  b)  Y  c  b)  z  Z(ca) 

III.  X(b  a)  V(bc)  z  Z(ca) 

IV.  X  .1  b   Y  be  .    /  1 

j     In  XY  .'  /.  X.  Y  and  /  may  each  take  on  the  tour  forms 
A.  i:.  1.  «». 

1  Prin.  i.     XY  l  VX 

Theorem  (XY  l  Z)  £  (YX  z  Z)  i  and  ii,  Ch.  V. 
We  may  consequently  confine  our  attention  to  one  order  <>t 
form--. 
I  )i  i~.  4.    The  term  occurring  in  both  prems.  called  middle  term. 
The  predicate  of  conclusion  called  major  term. 

The  SUbj.  of  Conclusion  called  minor  term. 

The  prem.  containing  major  term  called  major  premi 
The  prem.  containing  minor  term  called  minor  premise. 

B    convention  we  study  array  in  which  major  premise  is  written 

first. 

2  The  four  ways  in  which  the  terms  maj  be  ordered  yield 
four  figurt  11. 


50 


Letters  on  Logic  to  a 


By  convention  we  number  figures  in  the  order  given. 

(3)  In  each  figure  there  are  as  many  moods  as  there  are  com- 
binations of  the  four  forms  A,  E,  I,  O  taken  three  at  a  time 
(two  prems.  and  the  conclusion)  =  43  =  64  moods. 

The  array  is  then  constructed  under  each  fig.  as  follows: 
AA(A,E,  1,0)    EA(A,E,  1,0)    IA(A,E,I,0)    OA  (A,  E,  1,0) 

IE 

II 

10 


AE 

u 

EE 

AI 

a 

EI 

AO 

u 

EO 

OE 

01 

OO 


The  valid  moods. 


Prin.  ii.  Prin.  iii. 

(X  Y  z  Z)  z  (X  Z'  /  Y')  (XYzZ)z(Z'Yz  X') 

Prin.  iv.     (W  Z  X)  (X  Y  z  Z)  z  (W  Y  z  Z) 
Prin.  v.     (X  Y  z  Z)  (Z  z  W)  Z  (X  Y  z  W) 

Post.  1.     A(b  a)A(c  b)  z  A(c  a)     (A  A  A)i  is  valid  mood. 

Post.  2.     E(b  a)A(c  b)  z  E(c  a)     (E  A  E)x  is  valid  mood. 

Deduction  of  valid  moods. 

For  convenience  of  application  principles  of  deduction  may  be 
stated  in  form  of  following  rules: 

Rule  I.  Interchange  contradictories  of  either  premise  and  of 
the  conclusion. 

Rule  II.     Strengthen  premise  or  weaken  conclusion. 

Rule  III.     Convert  either  prem.  or  conclusion. 

Historical  Note:  The  problem  of  reduction. 


The  invalid 
Prin.  i. 
Prin.  ii. 
Theorem 
Theorem 
Post.  1 

Post.  2 

Post.  3 
Post.  4 
Post.  5 
Post.  6 
Post.  7 


(VII) 

Moods. 
(X  Y  z  Z)'  Z  (X  Z'  z  Y')' 

(X  Y  z  Z)'  Z  (Z'  Y  z  X')' 
.     (W  Z  X)  (W  Y  z  Z)'  Z  (X  Y  z  Z)' 
.     (X  Y  z  W)'  (Z  z  W)  Z  (X  Y  z  Z)' 

{ A(ba)  A(cb)  z  O(ca) } '  (AAO)i   is  an  invalid  mood 

{A(ba)E(cb)  z    I(ca)}'  (AEI)i     "  ' 

(A(ba)E(cb)  zO(ca))'  (AEOV 

(E(ba)E(cb)  z   I(ca)}'  (EEI),  " 

{0(ba)A(cb)  zO(ca)}'  (OAO)," 

(A(ab)A(bc)  zA(ca))'  (AAA)4" 

|A(ab)  A(bc)  z  0(ca)}'   (AAO)4" 


Y<n  NG    Man    \vi  rHOl  l     \    MASTER  S  ' 

Deduction  of  invalid  moods.  For  convenience  ol  application 
principles  of  deduction  may  be  stated  in  form  of  following  rul 

Rule  I.  Interchange  contradictories  of  either  premise  and  oi 
the  conclusion  of  invalid  mood. 

Rule  II.  Weaken  premise  or  strengthen  conclusion  of  invalid 
moot!. 

Rule   III.     Convert  either  premise  or  conclusion  of  invalid 

mood. 

Rules  for  immediate  detection  of  invalid  mood; 

Rule  I.  Two  negative  premises  do  not  imply  a  conclusion 
EEI)i. 

Rule  II.  Two  affirmative  form-  do  not  imply  a  negative 
conclusion  (AAO)* 

Rule  III.  An  affirmative  and  a  negative  premise  do  not 
imply  an  affirmative  conclusion  (AEI)i. 

Rule  IV.  Two  prems.  in  neither  of  which  middle  term  is 
distributed  <Io  not  imply  a  conclusion  (0  A  0)i. 

Rule  V.  Premises  in  which  a  given  term  is  undistributed  do 
not  imply  a  conclusion  in  which  that  term  is  distributed.  (AEO)i 
major  term;   (AAA)*  minor  term. 

Proof  of  the  necessity  d\u\  sufficiency  of  these  rules. 

(VIII) 

The  Sorites  (vid.  Ch.  VI,  Del".  3.) 

X:     [,  2    X      2,3       ■     X„  in.  n  +  1)  Z  Xn;,(n  +  I,  I) 

The  valid  mood-  ; 

Prin.  i.     (Y  z  Z)  z  (XV  z  XZ). 
Theorem:    If  X,d.  2)  X»(2,  3)  Z  X     J,  1 

X     ;.  1    X;  3,4     .    X     ,.  n 


X..      n.  .   X,..   ,(n,  n  +  1)  Z  X        n  +  I,  • 
Nun  X.     1.  2     X      2,  0  ...   X.      cn.n  +  i)  I   X.         n    •    1.  1    . 

The  invalid  mood-: 

Prin.  ii.  A  mood  is  invalid  if  the  mood  obtained  by  identifying 
any  two  term-  can  be  shown  to  be  invalid. 

Prin.  iii.  A  mood  containing  the  prem.  A(aa)  or  I  (aa)  is 
invalid  if  the  mood  from  which  this  prem.  is  omitted  can  be 
shown  to  be  in\  alid. 


52  Letters  ox  Logic  to  a 

Prin.  iv.  A  mood  containing  the  prems.  X(ab)X(ab)  is  invalid 
if  the  mood  containing  but  one  of  these  prems.  can  be  shown  to 
be  invalid. 

By  means  of  these  principles,  the  invalid  moods  of  sorites 
may  be  deduced  from  previous  results  without  further  postulates. 
We  find  that  the  valid  moods  constructed  under  Prin.  i  are  the 
only  valid  moods. 

(IX) 
The  Zero  cycle. 

Prin.  i.      (X  Z  Y)  Z  {XY'  Z  (XY')'}. 
Prin.  ii.      [(XY'  Z  (XY')'}   Z  (X  Z  Y). 
Def.  i.       {(XY')  z  (XY')7}  Z  (XY'  z  O). 
(XY'  zO)  z  {(XY')  z  (XY')'}. 

Def.  2.  When  the  antecedent  is  conjunctive  of  categorical 
forms  with  terms  in  cyclical  order  the  form  called  zero  cycle. 
Have  solved  zero  cycle  with  exception  of  form 

X(aa)  Z  O 
Post.  i.       0(aa)  Z  O 
Theor.  i.     E(aa)  Z  O 
Post.  2.        {A(aa)  z  O}' 
Theor.  2.     {  I(aa)  Z  O}' 


Voi  ng  Man  \\  1 1  ii<  'i  i    \  Masi  ef  5  | 


APPENDIX 

Note  on  nn-  Relation  oi   Subalternation 

The  relation  of  subalternation  being  all  but  universally  denied 

in  recent  times,  it  will  not  be  inappropriate  to  point  nut  in  what 
sense  this  denial  rests  upon  a  misapprehension.     The  following 

solution  is  due  to  Professor  Singer. 

If  we  employ  the  symbol,  z  ,  for  inclusion,  the  four  categi  u  ii 
forms  might  supposedly  be  represented  as  follows: 

(A)  All  a  is  b  =  (a  Z  b) 

(E)  No  a  is  b  =  (a  Z  b') 

(  I)  Some  a  is  b  =  (a  Z  b')' 

(O)  Some  a  is  not  b  =  (a  Z  b)' 

A  i-  now  the  contradictory  of  (>  and  E  is  the  contradictory  of 
I  but  A  Z  I  and  E  /  O  no  longer  hold  true. 

This  interpretation  of  Aristotle's  four  forms,  however,  is  in 
no  way  forced  upon  us,  for  we  may  assume : 

A  All  a  is  b  =  (a  Z  b) 

K  No  a  is  b  =  (a  Z  V) (a  Z  a')\b  Z  b')' 

(  I)  Some  a  is  b  =  (a  Z  b')'  +  (a  Z  a')  +  (b  Z  &0 

(O)  Some  a  is  not  b  =  (a  Z  &)' 

These  equalities  fulfill  the  essential  conditions: 

AE  z  0,        /  z  A   t.i  . 

the  lir-t  of  which  contains 

A  z  I  and  E  z  O. 

since  tlu-  members  of  tin-  pairs,  I  .  I  and  A.  <>  are  contradict* 
It  will  be  observed,  too,  that  E  and  I  retain  their  character! 
property  of  -imply  convertibility. 


I 


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